Rapid recovery of precise position after temporary signal loss

ABSTRACT

A real-time kinematic (RTK) filter uses the backup data to estimate a relative position vector between the mobile receiver at the first measurement time and the mobile receiver at the second measurement time and to provide recovery data associated with a satellite-differenced double-difference estimation for the mobile receiver between the first measurement time and the second measurement time. A navigation positioning estimator can apply the relative position vector, the backup data, the recovery data from the RTK filter, and received correction data with precise clock and orbit information on the satellite signals, as inputs, constraints, or both for convergence or resolution of wide-lane and narrow-lane ambiguities, and determination of a precise position, in accordance with a precise positioning algorithm.

RELATED APPLICATION

This document (including the drawings) claims priority and the benefitof the filing date based on U.S. provisional application No. 62/310,062,filed Mar. 18, 2016, under 35 U.S.C. § 119 (e), where the provisionalapplication is hereby incorporated by reference herein.

FIELD OF DISCLOSURE

This disclosure relates to a method and satellite receiver system forrapid recovery of precise position after temporary signal loss of one ormore satellite signals.

BACKGROUND

In certain prior art, satellite navigation receivers, such as GlobalNavigation Satellite System (GNSS) receivers, can provide centimeterlevel estimates of position. Such satellite navigation receivers oftenrely upon carrier phase measurements that are subject to integerambiguities of a cycle of the received satellite signal. In some priorart, long initialization periods, also known as pull-in times, aretypically 30 to 45 minutes, driven by the time it takes for phaseambiguities to converge to near stable values and for the solution toreach its optimal precision. Prior to the satellite navigation receiverresolving the integer ambiguities of the carrier phase measurements, theprecision of the position estimates is degraded.

In some prior art, real-time kinematic (RTK) correction data for thenavigation receiver is locally valid, rather than globally valid andrequires a significant investment in real-time kinematic base stationsand communications links to support communications between a basestation and a mobile satellite navigation receiver. RTK navigationapplications typically have been restricted to a short range of about 20kilometers to about 30 kilometers for single baseline between the basestation and rover because of distance-dependent biases between thereceiver and base station.

In other prior art, Precise Point Positioning (PPP) correction data isglobally valid and supports determination of an accurate positionsolution without regional corrections from additional regional referencestations. However, because PPP correction data only includes thesatellite-dependent portion of the measurement errors, prior art PPPmethods can take longer time than conventional RTK methods to reach fullpositioning accuracy.

Under certain circumstances, such as the presence of trees, buildings,obstructions, terrain height changes, fading, or an interfering signal,a GNSS system employing a prior art PPP method may experience a loss orinterruption of received (GNSS) satellite signal lock for short time(e.g., for a few minutes) and subsequently regain the signal after thebrief loss or interruption. In response to the loss of signal lock andafter the interruption, the prior art PPP estimator may reset such thata new convergence period (e.g., at least ten minutes and up to 30 to 45minutes) is required for the PPP-based-GNSS system to recover back tothe full positioning accuracy. Although some prior art attempts to usethe last available position estimate prior to loss of lock, theassumption that the last available position estimate did not changematerially during the interruption prior can be inaccurate. Accordingly,an operator of heavy equipment, agricultural equipment, constructionequipment, forestry equipment or other work vehicle may have significantwasted downtime waiting for the signal to converge, instead ofperforming work tasks with the work vehicle.

Thus, there is need for a method and satellite receiver system for rapidrecovery of precise position after temporary signal loss.

SUMMARY

In accordance with one embodiment, a mobile receiver is adapted toquickly or rapidly determine or recover a precise position based onhistorical data or backup data stored in a data storage device of themobile receiver. A receiver module can receive a set of one or moresatellite signals. A measurement module is capable of measuring thecarrier phase of one or more received satellite signals for a firstmeasurement time (e.g., t₁), a second measurement time (e.g., t₂), orboth. An estimator is adapted to estimate a wide-lane ambiguity andnarrow-lane ambiguity in the measured carrier phase of the one or morereceived satellite signals for the first measurement time and estimatingtropospheric bias for one or more of the carrier satellite signals basedon correction data. A data storage device is arranged to store, atregular time intervals for the first measurement time, backup datacomprising a set of any of the following post-convergence or resolvedvalues: the estimated wide-lane ambiguities (e.g., float or fixedvalues), fixed wide-lane ambiguities that are fixed to integer ambiguityvalues, the estimated narrow-lane ambiguities, fixed narrow-laneambiguities that are fixed to integer ambiguity values, resolvednarrow-lane ambiguities resolved to the combination of an integer valueand real numbered value for a satellite signal, the estimatedtropospheric delay bias, raw measured carrier phase of the receivedsatellite signals, and corresponding estimated receiver positions. Aloss or lack of reception is detected for one or more of the carriersignals for a loss time period. After the detected loss of lock once atleast some carrier phase signals are reacquired, the measurement moduleis adapted to measure the carrier phase of one or more receivedsatellite signals at a second measurement time. A real-time kinematic(RTK) filter uses the backup data to estimate a relative position vectorbetween the mobile receiver at the first measurement time and the mobilereceiver at the second measurement time and to provide recovery dataassociated with a satellite-differenced, double-difference measurementsfor the mobile receiver between the first measurement time and thesecond measurement time. A navigation positioning estimator can applythe relative position vector, the backup data, the recovery data fromthe RTK filter, and the correction data with precise clock and orbitinformation on the satellite signals, as inputs, constraints, or bothfor (rapid) convergence or (quick, efficient) resolution of wide-laneand narrow-lane ambiguities in accordance with a precise positioningalgorithm. The navigation positioning estimator is adapted to estimate aprecise position of the mobile receiver based on or derived from thehistorical resolved narrow-lane ambiguities and wide-lane ambiguitiesthat are in a converged state or fixed state, where the above isimplemented by a data processor of an electronic data processing systemof the mobile receiver.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a block diagram of one embodiment of a navigation satellitereceiver system for rapid determination of precise position by backupdata, where the navigation satellite receiver can obtain correction datafrom a communications satellite.

FIG. 1B is a block diagram of one embodiment of a navigation satellitereceiver system for rapid determination of precise position by backupdata, where the navigation satellite receiver can obtain correction datafrom a wireless communications network.

FIG. 2 is a block diagram of an illustrative example of satellitereceiver shown in greater detail than FIG. 1.

FIG. 3 illustrates one embodiment of method and satellite receiver foroperating a satellite receiver for rapid determination of preciseposition by backup data.

FIG. 4, which comprises FIG. 4A and FIG. 4B, collectively, illustratesanother embodiment of method for operating a satellite receiver forrapid determination of precise position by backup data.

FIG. 5 illustrates one embodiment of method for operating a satellitereceiver for rapid determination of precise position by backup data;and, more specifically, for testing constraints for reliability inarriving at a precise position.

DETAILED DESCRIPTION

A location-determining receiver, position-determining receiver, orsatellite receiver, such as a Global Navigation Satellite System (GNSS)receiver, is capable of receiving carrier phase measurements that aresubject to ambiguities, such as integer ambiguities, in the number ofcycles or fractional cycles of the received satellite signal. An epochor measurement time means a specific instant in time of a navigationsatellite system or the time interval during which the mobile receivermeasures the carrier phase (e.g., at a certain corresponding frequencyor rate). As used in this document, a mobile receiver is synonymous withthe term rover. The receiver determines or resolves ambiguities ofcarrier phase measurements to estimate accurately the precise positionor coordinates of the receiver. Although the code phase or pseudo-rangemeasurements of the GNSS receiver are not associated with integerambiguities in the cycles of the received satellite, code phasemeasurements do not provide the centimeter level position accuracyrequired for certain applications. As used throughout this document,ambiguities are often specific to the context of particular equations(e.g., later described in this document) which relate to observationsfrom one or more receivers of carrier phase signals from one or moresatellites. Accordingly, it is possible to have wide-lane (WL)ambiguities, narrow-lane (NL) ambiguities, single-difference (SD)ambiguities, double-difference (DD) ambiguities, real-time-kinematic(RTK) ambiguities, and refraction-corrected (RC) ambiguities that relateto phase measurements from one or more receivers, or one or moresatellites.

If the satellite navigation receiver can receive at least twofrequencies, such as L1 and L2 frequencies, the difference of the L1 andL2 carrier phase measurements can be combined to form wide-lane (WL)measurement (e.g., with a wavelength of approximately 86.25 centimetersfor GPS) and the sum of the L1 and L2 carrier phase measurements can becombined to form narrow-lane (NL) measurements (e.g., with a wavelengthof approximately 10.7 centimeters). The wide-lane measurementsfacilitate quick and efficient resolution of wide-lane integerambiguities, whereas the narrow lane measurements facilitate precise andaccurate resolution of narrow-lane ambiguities with minimal phase noise.The refraction-corrected ambiguities eliminate the first order ofionospheric delay.

Single-difference measurements (e.g., of carrier phase or code phase)can be formed between two receivers and a satellite. For example, singledifference measurements can be formed between a reference receiver 130(at a known location) and one satellite and between a rover receiver andthe satellite (e.g., to eliminate or reduce certain errors that arecommon to both receivers). Single difference measurements (of carrierphase or code phase) can be formed with respect to one receiver (e.g.,reference receiver 130 or rover) and a pair of satellites (e.g., at thesame observation time to reduce or eliminate receiver clock error).

Double-difference measurements can be formed by subtracting two relatedsingle-difference measurements. For example, double-differencemeasurements (e.g., of carrier phase or code phase) can be formed withrespect to one satellite, a reference receiver 130 and a rover receiver20 by subtracting two single difference measurements. Further,double-difference measurements can be formed with respect to twosatellites and a rover receiver (e.g., 20), or by subtracting twosingle-difference measurement. In certain embodiments, differences maybe taken at the same observation times or different observation timesand for different frequencies or combinations of received satellitesignals.

As used herein, ambiguities that are estimated, determined or “resolved”may have integer values, float values or real number values.Accordingly, estimated ambiguities, determined ambiguities and resolvedambiguities shall be regarded as synonymous terms in this document. Incontrast, ambiguities that are “fixed” shall mean the ambiguities haveinteger values, unless otherwise specified, such as where ambiguitiesare divided into a fixed integer component and a real value component(float component). Converged ambiguities refer to integer or real valuedambiguities that are associated with reliable or steady-state accuratesolutions or position estimates that are at or approach peak accuracyand acceptable standard deviation levels for a GNSS receiver operatingin a GNSS.

The measurement module 56 or navigation receiver (e.g., 20) can measureor observe the L1 and L2 carrier phases and pseudo-ranges of theapplicable Global Navigation Satellite System (GNSS) (e.g., GlobalPositioning System (GPS) or GLONASS) as shown in Equations (1-4) asfollows:

$\begin{matrix}{\mspace{79mu}{P_{1}^{j} = {\rho^{j} + \tau_{r} + \tau^{j} + T + b_{P_{1}} + B_{P_{1}}^{j} + I^{j} + ɛ_{P_{1}}^{j}}}} & (1) \\{\mspace{79mu}{P_{2}^{j} = {\rho^{j} + \tau_{r} + \tau^{j} + T + b_{P_{2}} + B_{P_{2}}^{j} + {\frac{f_{1}^{2}}{f_{2}^{2}}I^{j}} + ɛ_{P_{2}}^{j}}}} & (2) \\{L_{1}^{j} = {{\Phi_{1}^{j}\lambda_{1}} = {\rho^{j} + \tau_{r} + \tau^{j} + T + b_{L_{1}} + B_{L_{1}}^{j} - I^{j} + {N_{1}^{j}\lambda_{1}} + {( {W^{j} + w} )\lambda_{1}} + ɛ_{L_{1}}^{j}}}} & (3) \\{L_{2}^{j} = {{\Phi_{2}^{j}\lambda_{2}} = {\rho^{j} + \tau_{r} + \tau^{j} + T + b_{L_{2}} + B_{L_{2}}^{j} - {\frac{f_{1}^{2}}{f_{2}^{2}}I^{j}} + {N_{2}^{j}\lambda_{2}} + {( {W^{j} + w} )\lambda_{2}} + ɛ_{L_{2}}^{j}}}} & (4)\end{matrix}$where:

-   -   P_(i) ^(j) and L_(i) ^(j) are pseudo-range and carrier phase        measurements (e.g., in meters), respectively, for a given        frequency i (e.g., 1, 2, . . . , such as L1 or L2) and satellite        j;    -   Φ₁ ^(j) is an ambiguous or non-integer phase measurement and λ₁        is the wavelength of the carrier phase measurement for frequency        L1;    -   Φ₂ ^(j) is an ambiguous or non-integer phase measurement and λ₂        is the wavelength of the carrier phase measurement for frequency        L2;    -   ρ^(j) is the geometric distance (e.g., in meters) between the        satellite j phase center and the receiver phase center including        satellite orbital correction in the correction data 108,        receiver tide displacement and earth rotation correction;    -   τ_(r) is the receiver r clock bias or error for a given GNSS        system, where one receiver clock bias is estimated for each GNSS        system such as GPS, GLONASS, Galileo or Beidou constellation;    -   τ^(j) is the satellite clock error for satellite j;    -   T is the tropospheric delay, and is divided into a dry component        T_(dry) and a wet component T_(wet);    -   b_(P) _(i) and b_(L) _(i) are receiver dependent code bias and        phase bias, respectively, for a given frequency i (1, 2, . . . )        and can be assumed to be same for each CDMA signal of all the        visible satellites within each GNSS constellation;    -   B_(P) _(i) ^(j) and B_(L) _(i) ^(j) are satellite j dependent        code bias and phase bias, respectively, for a given frequency i        (1, 2, . . . ) which change very slowly over time;    -   f_(i) and λ_(i) are the GNSS carrier signal frequency i and its        wavelength;    -   I^(j) is the ionosphere error for a given satellite j;    -   N_(i) ^(j) is carrier phase integer ambiguity for a given        frequency i and satellite j;    -   W^(j) and w are phase windup errors for both satellite j and        receiver, in cycles, respectively, which can be corrected with        models;    -   ε_(P) _(i) ^(j) and ε_(L) _(i) ^(j) are code and phase errors,        respectively, including white noise, multipath and remaining        model errors for satellite j and frequency i.        In an alternate embodiment, an alternative approach for the        receiver r clock bias, τ_(r), is the receiver r clock bias is to        estimate one clock for a primary constellation such as GPS and        then relative receiver clock biases between primary        constellation and the other GNSS constellations.

For determination of the tropospheric delay, the dry component can beaccurately modeled using an a priori troposphere model, such as theGlobal Pressure and Temperature model (GPT) or the GPT2 model; theremaining wet component, after removing an a priori wet model, can befurther estimated as one zenith bias with elevation mapping functionand/or additional two horizontal gradient coefficients.

If the measurement module 56 or receiver (e.g., 20) observes or measuresGLONASS satellite signals, the different frequencies of differentsatellite transmitters 100 must be considered. For example, thesatellite signals transmitted by GLONASS satellites can be derived froma fundamental frequency (1602 MHz for L1 band, 1246 MHz for L2 band) ofthe satellite L-Band. Each GLONASS satellite currently transmits on adifferent frequency using an FDMA technique. The equation to give theexact L1 center frequency is as follows:f ₁ ^(j)=1602 MHz+n ^(j)×0.5625 MHz  (5)where n^(j) is frequency channel number (n=−7, −6, . . . , 6) ofsatellite j. On the L2 band, the center frequency is determined by theequationf ₂ ^(j)=1246 MHz+n ^(j)×0.4375 MHz  (6)

FIG. 1A is a block diagram of one embodiment of a satellite receiversystem 11 for rapid determination of precise position by correction data108 received wirelessly from a correction data source 24, backup datastored in a data storage device 62 (in FIG. 2) associated with themobile receiver 20, and recovery data generated by relative positioningmodule 18 (in FIG. 2) or real-time kinematic filter associated with themobile receiver 20. A correction data source 24 transmits correctiondata 108 via a wireless signal to the mobile receiver 20 or rover andthe correction data 108 is received via a correction wireless device 26associated with the mobile receiver 20.

In one embodiment, the correction data source 24 comprises an electronicsystem for generation and distribution of correction data 108. Asillustrated in FIG. 1A, the correction data source 24 comprises a(global) reference data network 132, a data processing center 118, aterrestrial uplink station 128 and a communications satellite 135.

In one embodiment, reference receiver 130 measures the carrier phase ofone or more of the carrier signals or received satellite signals from aset of satellite transmitters 100 on satellites orbiting the Earth. Thereference receiver 130 can also measure the pseudo-range or code phaseof a pseudo-random noise code that is encoded on one or more of thecarrier signals from the set of satellite transmitters 100. Thereference receivers 130 receive and send measurements, ephemeris data,other observables and any information derived from the deliverables toan electronic data processing center 118 (e.g., hub). In one embodiment,each reference receiver 130 transmits (e.g., via a communications link,a communications network, a wireless channel, a communications channel,communications line, a transmission line, or otherwise) a set of carrierphase measurements of received satellite signals, and associatedsatellite identifiers, and ephemeris data to an electronic dataprocessing center 118 (e.g., reference data processing hub).

The data processing center 118 or its correction data estimator 134determines correction data 108 (e.g., precise correction data) in realtime based on the measurements, ephemeris data, other observables andany derived information received from one or more reference receivers130. In one embodiment, the data processing center 118 comprises anelectronic data processor 120, a data storage device 124, and one ormore data ports 126 that are coupled to a data bus 122. The dataprocessor 120, the data storage device 124 and the one or more dataports 126 may communicate with each other via the data bus 122.

Software instructions and data that are stored in the data storagedevice 124 may be executed by the data processor 120 to implement any ofthe blocks, components or modules (e.g., electronic modules, softwaremodules, or both) described in this disclosure document. The dataprocessor 120 may comprise a microcontroller, a microprocessor, aprogrammable logic array, an application specific integrated circuit(ASIC), a digital signal processor, or another device for processingdata, manipulating, accessing, retrieving, and storing data. A datastorage device 124 may comprise electronic member, non-volatileelectronic memory, an optical storage device, a magnetic storage device,or another device for storing digital or analog data on a tangiblestorage medium, such as an optical disk, a magnetic disk, or electronicmemory. Each data port 126 may comprise a buffer memory, a transceiveror both for interfacing with other network elements, such as a referencereceiver 130 or a terrestrial satellite uplink station 128.

In one embodiment, the data processing center 118 or data processor 120or correction data estimator 134 receives the phase measurements andcorresponding satellite identifiers from the reference receivers 130,reference receiver identifiers (or corresponding coordinates) andprocesses the phase measurements to estimate a clock bias for eachsatellite, or rather each satellite signal, or a corresponding clocksolution for incorporation into correction data 108. As illustrated inFIG. 1A, the clock solution, clock bias or correction data 108 isprovided to a terrestrial uplink station 128 or another communicationslink. For example, the terrestrial uplink station 128 communicates ortransmits the clock solution, clock biases or correction data 108 to acommunications satellite 135 (e.g., repeater).

In turn, the communications satellite 135 transmits the correction data108 to a correction wireless device 26 (e.g., a satellite receiver orL-band satellite receiver) at a mobile receiver 20. The correctionwireless device 26 is coupled to a mobile receiver 20 (e.g., mobile GNSSreceiver) or rover. The mobile receiver 20 also receives satellitesignals from one or more GNSS satellites and measures the carrier phase(and code phase) of the received satellite signals. In conjunction withthe phase measurements the precise orbit correction data, clockcorrection data, satellite wide-lane bias and satellite narrow-lane bias(e.g., for each satellite) in the correction data 108 can be used toestimate the precise position, attitude, or velocity (e.g., solution) ofthe mobile receiver 20, or its antenna. For example, the mobile receiver20 may employ a precise point positioning (PPP) estimate using preciseclock and orbital solutions for the received signals of the satellites.

The system 111 of FIG. 1B is similar to the system 11 of FIG. 1A exceptthe system of FIG. 1B replaces the communications satellite 135 and theterrestrial uplink station 128 with a communications device 127 (e.g.,server), a communications network 139 (e.g., Internet or communicationslink), and a wireless communications system 235. Like reference numbersin FIG. 1A and FIG. 1B indicate like elements, modules or features.

As illustrated in FIG. 1B, the correction data source 124 comprises areference data network 132, a data processing center 118, acommunications device 127 and a wireless communications system 235. Inone embodiment, the wireless communications system 235 may comprise acellular communications system, a trunking system, a WiFi communicationssystem, or another communications system. For example, the cellularcommunications system may comprise cell sites or base stations incommunication with a base station controller, a router, or anothermobile telephone switching office (MTSO), where the MTSO interfaces witha communications network 139, such as the Internet.

The communications network 139 may comprise microwave links, fiberoptical links, the public switched telephone network (PSTN), theInternet, or another electronic communications network. In oneembodiment, the communications device 127 comprises a server thatformats, organizes or transmits the correction data in data packets(e.g., data packets compatible with TCP/IP Transmission ControlProtocol/Internet Protocol) for transmission over the communicationsnetwork 139. The communications network 139 communicates with thecorrection wireless device 226 (e.g., cellular transceiver) that isassociated with or coupled to the mobile receiver 20.

In this document, under the precise positioning mode of FIG. 1A or FIG.1B, the mobile receiver 20 can achieve centimeter-level accuracypositioning, by using the real-time global differential correction data108. This correction data 108 is available and valid globally througheither over satellite communications (e.g., L-Band geostationarycommunication satellite) in FIG. 1A or wireless communications system(e.g., cellular wireless system) in FIG. 1B. The global differentialcorrection under a precise positioning mode, illustrated in the exampleof FIG. 1A or FIG. 1B, eliminates the need for local reference stationsand radio communication that would otherwise be used to establish shortbaselines (e.g., less than approximately 20 kilometers to approximately30 kilometers) between a reference receiver 130 and a mobile receiver 20for precise position accuracy.

The correction data 108 may comprise precise orbit and clock correctionsand any other satellite bias data that is necessary or useful to providea precise point position (PPP) data services (e.g., withcentimeter-level accuracy) to position-determining receivers in one ormore geographic regions or throughout the world. The correction data 108with additional satellite bias data enable mobile receivers 22 toquickly converge and pull-in to precise accuracy (e.g., centimeter levelaccuracy) or peak accuracy levels.

From time to time, any mobile receiver 20 may experience signalinterruption of one or more satellite signals (from one or moresatellite transmitters 100) for various reasons, such as signalpropagation variation, electromagnetic interference, electromagneticnoise, signal attenuation, signal fading, multipath signal reception,tree obstructions, vegetation obstructions, terrain obstructions,building obstructions, the setting of a satellite from view or receptionrange, among other reasons. Obstructions refer to structures or objectsthat can attenuate or block propagation of the satellite signals betweenany satellite transmitter 100 and the mobile receiver 20. If a mobilereceiver 20 or method of this disclosure experiences a temporary loss orinterruption of one or more received satellite signals (e.g., GNSSsignals) for short time (e.g., for a few minutes) and subsequentlyregains one or more of the received satellite signal after the briefloss or interruption, the receiver or method can employ an innovativetechnique of recovering the converged position solution (e.g., PPPsolution or position estimate) rapidly to approximately the same levelof accuracy before the signal interruption by estimating the differencesof navigation states, such as position, carrier phase ambiguities, orthe like. The above innovative technique may be referred to as a rapidrecovery (RR) technique. In one embodiment, the rapid recovery techniqueallows a receiver to recover from the time gap in proper phasemeasurements of the received satellite signals, where the time gap isless than or equal to a maximum time period. For example, for a time gapor signal interruption of several minutes, the method or receiver 20 canrecover a precise position estimate of the receiver almost immediately(e.g., within some seconds) based on reference to stored historicaldata, once the receiver resumes tracking of received satellite signalsand generating phase measurements of the received satellite signalsfollowing the blockage event.

In this disclosure, the precise point positioning (PPP) algorithm canprovide centimeter level accuracy, using the correction data 108, whichincludes a single set of clock and orbit corrections with globalvalidity, generated from a sparse global network of reference stations.Unlike certain prior art real-time kinematic systems for providingcorrection data 108, PPP eliminates the need for a dense network ofreal-time kinematic (RTK) base stations and associated wirelesscommunications links to support determination of correction data 108with local validity or communications of a correction signal between anRTK base station and rover 200.

As illustrated in FIG. 1A and FIG. 1B, a reference position-determiningreceiver or reference receiver 130 receives satellite signals from afirst set of satellites (e.g., satellite transmitters 100) within viewor reliable reception range. In one configuration, a received satellitesignal has a carrier signal that is encoded with a pseudo-random noisecode or other spread-spectrum code.

The mobile position-determining receiver or rover 20 receives satellitesignals from a second set of satellites (e.g., satellite transmitters100) within view or reliable reception range. As used in this document,the terms rover 20 and mobile receiver 20 shall be synonymous. For theRTK algorithm executed by the relative positioning module 18, thereneeds to be commonality between the member satellites in the first andsecond sets of satellites within view or reliable range of the referencereceiver 130 and the rover 20. However, for the PPP algorithm executedby the precise positioning module 16 in the rover 20, the rover 20 mayuse additional satellites that are not within the first set.

For the relative positioning module 18 or real-time kinematic (RTK)filter 48 to provide accurate results from recovery from a signalinterruption, the mobile receiver 20 needs to be within a maximum rangeor distance from a first measurement time (e.g., t₁) to a secondmeasurement time (e.g., t₂). For example, if the mobile receiver 20 hasnot moved by more than a maximum of range of zero to approximatelythirty kilometers from the first measurement time to the secondmeasurement time, the double difference equations used by the real-timekinematic filter 48 can provide accurate results. In alternateembodiments, if the mobile receiver 20 has not moved by more than amaximum of range of zero to approximately fifty kilometers from thefirst measurement time to the second measurement time, the real-timekinematic filter 48 can provide acceptable or adequate results.Accordingly, the relative positioning module 18 or real-time kinematicfilter 48 may verify that the mobile receiver 20 has not moved by morethan a range of zero to approximately thirty kilometers between thefirst measurement time and the second measurement time prior to applyingthe real-time kinematic (RTK) filter 48 to resolve ambiguitiesassociated with double-difference carrier phase measurements.

The rover 20 is coupled to or in communication with a correctionwireless device 26 that receives correction data 108 from the correctiondata source (24, 124) via the transmitted wireless correction signal.

As illustrated in FIG. 1A and FIG. 1B, the mobile receiver 20 attains aconverged state with final precise accuracy after resolving ambiguitiesassociated with the carrier phase signals based on the receivedcorrection data 108 in accordance with a precise point positionalgorithm or a precise positioning module 16.

In an alternate embodiment, the mobile receiver 20 attains a convergedstate with final precise accuracy at a first measurement time (e.g., t₁)after resolving ambiguities associated with the carrier phase signalsbased on the received correction data 108 in accordance with a real-timekinematic (RTK) positioning algorithm or other differential correctionalgorithm.

At any time after the mobile receiver 20 achieves the converged state,for ambiguity resolution of the carrier phase, the precise positioningmodule 16 or the navigation positioning estimator 50 stores one or moreof the following backup data (e.g., including the resolved or convergedambiguity content) in the data storage device 62 associated with themobile receiver 20: resolved wide-lane ambiguities (e.g., fixed integersor float, real-valued numbers); fixed wide-lane ambiguities that arefixed to integer ambiguity values; resolved narrow-lane ambiguities orresolved refraction-corrected ambiguities; estimated narrow-laneambiguities (e.g., fixed integers or float, real valued numbers); fixednarrow-lane ambiguities that are fixed to integer ambiguity values;resolved narrow-lane ambiguities resolved to the combination of aninteger value and real numbered value for a satellite signal, estimatedtropospheric delay bias (e.g., residual tropospheric bias, troposphericdelay at zenith direction including the a priori model), rawmeasurements (e.g., phase or pseudo-range measurements), raw measuredcarrier phase, measured code phase of the received satellite signals,and mobile receiver position at the first measurement time (e.g., t₁).Generally, the resolved ambiguities comprise the pulled-in wide-lane andnarrow-lane ambiguities from one or more GNSS satellites' carriersignals the absolute tropospheric delay at zenith direction including apriori model and residual tropospheric delay estimates. The pulled-inambiguities are associated with position estimates that achievesteady-state accuracy approaching a peak precise accuracy, such asposition estimates (e.g., less than five centimeters horizontal orpass-to-pass position accuracy) within acceptable standard deviationmetrics (e.g., within one standard deviation) for a target percentagetime reliability or availability (e.g., approximately 95 percentreliability).

At the mobile receiver 20 or rover, the relative positioning module 18or the real-time kinematic filter 48 retrieves or reads the backup dataafter the temporary signal interruption or after the reception of one ormore received satellite signals is restored; the relative positioningmodule 18 applies a real-time kinematic (RTK) algorithm to providerelative position vector between reference receiver 130 between thefirst measurement time and the second measurement time and recoverydata. For example, the relative positioning module 18 usesdouble-difference of phase measurements between the reference receiver130 and the rover 20 and two satellites to resolve double-difference RTKambiguities, or related data, that are used as recovery data. At therover 20, the precise positioning module 16 applies the relativeposition vector, the backup data, recovery data, and correction data 108as inputs, constraints, or both for convergence of one or morepredictive filters (38, 40, 44) on wide-lane and narrow-lane ambiguities(e.g., in accordance with a precise positioning algorithm). In oneexample, the recovery data comprises L1/L2 fixed double-difference (DD)ambiguities from the real-time kinematic filter 48 at the mobilereceiver 20 based on raw phase measurements at the mobile receiver 20and the reference receiver 130 for a respective pair of satellites. Atthe rover 20, the precise positioning module 16 or the navigationpositioning estimator 50 estimates a precise position of the rover 20based on or derived from the converged or fixed narrow-lane ambiguitiesand wide-lane ambiguities.

FIG. 2 is a block diagram of an illustrative example of satellitereceiver shown in greater detail than FIG. 1. The position determiningreceiver of FIG. 2 may be used as a reference receiver 130, a rover 20or both.

In one embodiment, the mobile receiver 20 comprises a receiver front end10 coupled to an electronic data processing system 152. The receiverfront end 10 comprises an antenna, a radio frequency (RF) front end 12,and an analog-to-digital (A/D) converter 14. The RF front end 12 mayinclude one or more of the following: an radio frequency amplifier ormicrowave amplifier, a filter (e.g., bandpass filter), and adownconverter for down-converting the received satellite signal to anintermediate frequency signal or a baseband signal.

The electronic data processing system 152 includes that portion of thereceiver that processes data after the analog-to-digital conversion bythe analog-to-digital converter 14. For example, the electronic dataprocessing system 152 can comprise an electronic data processor 66, adata storage device 62 (e.g., electronic memory) and a data bus 64 forcommunication between the electronic data processor 66 and the datastorage device 62, where software instructions and data are stored inthe data storage device 62 and executed by the data processor 66 toimplement any of the blocks, components or modules (e.g., electronicmodules, software modules, or both) illustrated in FIG. 2. The mobilereceiver 20 may comprise a location-determining receiver for: (a)determining a location or precise position (e.g. three-dimensionalcoordinates) of a receiver antenna, (b) a range-determining receiver fordetermining a range or distance between the receiver antenna and asatellite (e.g., satellite antenna) or (c) determining ranges betweenthe receiver antenna and one or more satellites, or (d) determiningposition, velocity, acceleration, or attitude (e.g., yaw, pitch, roll)of the mobile receiver 20 or its antenna.

The analog-to-digital converter 14 converts the analog intermediatefrequency signal or analog baseband signal to a digital signal. Thedigital signal comprises one or more digital samples that are availableat a sampling rate. Each sample has a finite quantization level and eachsample is capable of being processed by an electronic data processingsystem 152.

In one embodiment, the data storage device 62 stores the followingmodules or components: baseband/intermediate frequency processing module54, measurement module 56, and navigation positioning estimator 50.

The baseband/intermediate frequency (IF) processing module 54 ormeasurement module 56 processes the digital signals. The measurementmodule 56 or a carrier phase measurement module 58 measures or detectsthe carrier phase of the received satellite signals from a set of GNSSsatellites with view or reception range. For example, the measurementmodule 56 measures the carrier phase of the received signal bycorrelating the received digital signal to a locally generated referencesignal. However, the measurement module 56 or carrier phase measurementmodule 58 measures the carrier phase of the satellite signals subject toan ambiguity or integer ambiguity in the number of cycles in any pathbetween the receiver antenna and the satellite. The measurement module56 or the code phase measurement module 60 measures the code phase orpseudo-range of the received satellite signals.

In one configuration, the measurement module 56 further comprises anoptional cycle slip detector 59 that detects a cycle slip or loss ofcontinuity in the tracking of the carrier phase of the received carriersignal from one or more satellites. For each satellite signal where acycle slip is detected (e.g., by detector 59) and the received satellitesignal is reacquired within a maximum time period, the rapid recoveryprocess of this disclosure can be used to recover the ambiguity orrapidly converge on the new ambiguity, rather than restarting theambiguity resolution process from scratch for that satellite signal andignoring the received satellite signal until floating ambiguity or fixedinteger ambiguity is reached. The optional nature of the cycle slipmodule 59 are indicated by its dashed lines.

The baseband/intermediate frequency processing module 54 is coupled to,or in communication with, the navigation positioning estimator 50. Inone embodiment, the navigation positioning estimator 50 comprises aprecise positioning module 16 (e.g., precise point positioning (PPP)module) and a relative positioning module 18.

In certain embodiments, the precise positioning module 16 represents aPPP estimator. The precise positioning module 16 can execute a precisepoint positioning algorithm to estimate a precise position of thereceiver or its antenna based on received correction data 108 via thecorrection wireless device 26. In general, in one embodiment the precisepositioning module 16 comprises a predictive filter, such as a Kalmanfilter or modified Kalman filter.

In one embodiment, the precise positioning module 16 may comprise anoptional zero-difference filter 38, a wide-lane filter 40, a narrow-lanefilter 44, a backup/recovery module 46, and an optional atmospheric biasestimator 42. The zero-difference filter 38 and the atmospheric biasestimator 42 are indicated as optional by the dashed lines in FIG. 2.Although the zero-different filter 38 may comprise a wide-lane filter 40and a narrow-lane filter 44 as illustrated, the precise positioningmodule 16 can realize one or more single-difference filters ordouble-difference filters for the wide-lane ambiguity resolution,narrow-lane ambiguity resolution, or resolution of ionosphere-freeambiguities.

In one embodiment, the precise positioning module 16 comprises a precisepoint positioning module that operates in accordance with a precisepoint positioning algorithm. For illustrative purposes, the followingequations can be used to implement one possible embodiment as follows.

The observation model that has been widely used for PPP is based onionosphere-free code and carrier phase observations that eliminate thefirst order of ionosphere error as shown in Equations (1-4). Theobservations, such as carrier phase and code phase measurements,received from all the satellites are processed together in one or morepredictive filters (e.g., Kalman filters, or the combination of awide-lane filter 40 and a narrow-lane filter 44) that solves for thedifferent unknowns, namely the receiver coordinates, the receiver clock,the zenith tropospheric delay and the phase floating ambiguities. Theaccuracy of the satellite clocks and orbits is one of the most importantfactors affecting the quality of the PPP solution. In order to achieveits full potential to applications, PPP faces two major challengesincluding a long initialization time and robust and reliable integerambiguity resolution to derive a more precise solution.

In one embodiment, the wide-lane filter 40, which can be applied to PPPdetermination, uses the following equations described below. Given thecode and phase measurements from two frequencies, such as L1 and L2 forGPS, G1 and G2 for GLONASS, the Melbourne-Wübbena linear combinationL_(WL) ^(j) can be formed as shown below.

$\begin{matrix}{L_{WL}^{j} = {( {{\frac{f_{1}}{f_{1} + f_{2}}P_{1}^{j}} + {\frac{f_{2}}{f_{1} + f_{2}}P_{2}^{j}}} ) - ( {{\frac{f_{1}}{f_{1} - f_{2}}L_{1}^{j}} - {\frac{f_{2}}{f_{1} - f_{2}}L_{2}^{j}}} )}} & (7)\end{matrix}$By expanding the above equation (7) using Equations (1)-(4), it can beshown that the geometric range related terms, which include range,receiver and satellite clock, ionosphere and troposphere errors and thephase wind-up term, are cancelled. It can be expressed in Equation (8)asL _(WL) ^(j) =N _(WL) ^(j)λ_(WL) +b _(WL) +B _(WL) ^(j)+IFB^(j)+ε_(WL)^(j)  (8)where:

-   -   λ_(WL) is wide-lane wavelength, about 86.4 cm for GPS and c is        speed of light,

$\begin{matrix}{{\lambda_{WL} = \frac{c}{f_{1} - f_{2}}};} & (9)\end{matrix}$

-   -   N_(WL) ^(j) is integer wide-lane ambiguity for satellite j,        N _(WL) ^(j) =N ₁ ^(j) −N ₂ ^(j);  (1)    -   where b_(WL) is wide-lane receiver bias (one per receiver and        constellation for all visible satellites), which is a        combination of L1 and L2 receiver code bias and phase bias, as        indicated in Equation (11):

$\begin{matrix}{{b_{WL} = {( {{\frac{f_{1}}{f_{1} + f_{2}}b_{P_{1}}} + {\frac{f_{2}}{f_{1} + f_{2}}b_{P_{1}}}} ) - ( {{\frac{f_{1}}{f_{1} - f_{2}}b_{L_{1}}} - {\frac{f_{2}}{f_{1} - f_{2}}b_{L_{2}}}} )}},} & (11)\end{matrix}$

-   -   -   where majority of GLONASS inter-frequency bias b_(P) ₁ and            b_(P) ₂ in code measurement is usually assumed to be linear            or a trigonometric function of the GLONASS satellite            frequency number; it is not the same for all the visible            satellite as with the case of CDMA signals such as GPS;        -   where IFB^(j) is the inter-frequency bias for satellite j,            such as for a GLONASS satellite;        -   where B_(WL) ^(j) is wide-lane satellite j bias (one per            satellite); and        -   where ε_(WL) ^(j) is the wide lane measurement error for            satellite j including white noise, multipath and remaining            un-modeled errors.

With respect to the inter-frequency bias per satellite, the linear modelcan be approximated below for GLONASS constellation as Equation (12):IFB^(j) ≈k·n ^(j)  (12)

-   -   where k is the IFB coefficient for receiver code bias. The IFB        varies from receiver to receiver, also varies from one siting        (antenna and cabling setup) to another siting. Modelled in this        way, typically k is less than 0.1.

The wide-lane satellite j bias, B_(WL) ^(j), (one per satellite) is acombination of L1 and L2 satellite code bias and satellite phase bias asin Equation (13); the satellite bias is changing slowly over time; bothsatellite and receiver wide-lane biases are not constant over time:

$\begin{matrix}{B_{WL}^{j} = {{- ( {{\frac{f_{1}}{f_{1} + f_{2}}B_{P_{1}}^{j}} + {\frac{f_{2}}{f_{1} + f_{2}}B_{P_{2}}^{j}}} )} + ( {{\frac{f_{1}}{f_{1} - f_{2}}B_{L_{1}}^{j}} - {\frac{f_{2}}{f_{1} - f_{2}}B_{L_{2}}^{j}}} )}} & (13)\end{matrix}$

-   -   where B_(P) ₁ ^(j) is satellite bias for satellite j of the code        phase or pseudo-range signal encoded on frequency L1 (f₁), where        B_(P) ₂ ^(j) is satellite bias for satellite j of the code phase        or pseudo-range on frequency L2 (f₂), where B_(L) ₁ ^(j) is        satellite bias for satellite j of the carrier phase on frequency        L1, where B_(L) ₂ ^(j) is satellite bias for satellite j of the        carrier code on frequency L2.

An optional zero difference filter can be used to determineundifferenced or zero-differenced (ZD) ambiguity states or floatambiguity states associated with the carrier phase measurements of thereceived satellite signals. The zero difference filter 38 is illustratedin dashed lines in FIG. 2 to show that the zero difference filter 38 isoptional and may be included within the wide-lane filter 40 in alternateembodiments. For example, the zero-differenced ambiguity state can bedetermined based on correction data 108 that contains satellite biasinformation from a network or group of reference receivers 20.

The wide-lane filter 40 uses zero-differenced (ZD) Melbourne-Wübbenalinear combination L_(WL) ^(j) in Eq. (7) as the input measurement toestimate one wide-lane floating ambiguity state N_(WL) ^(j) per visiblesatellite. The wide-lane satellite bias B_(WL) ^(j) can be broadcast inreal-time within correction data 108 or correction signals to mobilereceivers and will compensate for that term using Equation (8).

The precise positioning module 16 or wide-lane filter 40 lumps thereceiver wide lane bias b_(WL) into float WL ambiguity state N_(WL)^(j). Accordingly, the ZD WL ambiguity does not hold an integercharacteristic because all of the ZD WL ambiguities include the commonreceiver wide lane bias. However, the single differenced (SD) wide-laneambiguities between satellites within each constellation (e.g., GPSconstellation) at a mobile receiver or reference receiver are stillintegers and can be resolved in accordance with SD equations. Further,DD narrow-lane ambiguities, the DD wide-lane ambiguities, or the DDL1/L2 ambiguities between measurement times (or between epochs) withineach constellation are still integers and can be resolved in accordancewith double difference equations formed by subtraction of two SDobservations with the benefit of receiver bias cancellation. For theGLONASS constellation, the additional inter-frequency bias (IFB) statemay be required in order to preserve the integer nature of the SDambiguities.

Given that the actual ZD float ambiguity state variable is the sum of ZDinteger ambiguity and receiver bias, as explained above, dynamic updatefor the receiver bias variance needs to be included in the process noisemodel for the ZD ambiguity states as shown below in Equation 14 asfollows:

$\begin{matrix}{{Q_{WL}(t)} = {{Q_{WL}( {t - 1} )} + {\begin{pmatrix}1 & \ldots & 1 \\\vdots & \ddots & \vdots \\1 & \ldots & 1\end{pmatrix}{q_{b_{WL}} \cdot \Delta}\; t}}} & (2)\end{matrix}$where Q_(WL) is the time-varying receiver bias variance for thewide-lane ambiguities, q_(bWL) is the process noise associated with amatrix of ones or all-ones matrix, and Δt is the time interval betweentime t−1 and t.

Equation (8) will be used for wide-lane filter 40. The zero differencingwide-lane raw observation is used for the measurement update in thewide-lane filter 40. The state variables include one float wideambiguity per visible satellite, each conceptually including thewide-lane integer ambiguity and common receiver phase bias for therespective constellation.

In the navigation positioning estimator 50 of the mobile receiver 20,the wide-lane filter 40 will begin processing even before the satellitewide-lane (WL) bias correction from the correction data 108 is receivedor even if the satellite WL bias correction is invalid. The floatwide-lane (WL) ambiguity is reduced by the satellite wide-lane biascorrection when the satellite WL bias correction becomes valid (e.g.,makes a transition to valid from invalid states). Likewise, thesatellite WL bias is removed from the float wide-lane ambiguity (and thesatellite WL bias is increased) when the satellite wide-lane biasbecomes invalid (e.g., makes a transition from valid to invalid states).

In one embodiment, in the navigation positioning estimator 50 or thewide lane filter 40, the float ambiguity will be adjusted whenever +/−2cycle jumps of the satellite wide-lane (WL) bias are detected, whichindicates a transition between valid and invalid states. The aboveadjustment of the satellite wide-lane bias is limited to deviations of+/−2 cycles deviations to reduce the bandwidth or resources required fordata processing. As described above, the between-satellite singledifferencing ambiguity resolution for each constellation can beconducted, which is the equivalent of double-differencing ambiguityresolution. The satellites without a valid satellite wide-lane bias willbe skipped by the ambiguity resolution process, and the covariancematrix term may be inflated by a term representing small variance suchas 1^(e-4) cycles-squared once the corresponding ambiguity has beenfixed.

As previously suggested, the wide-lane filter 40 may comprise azero-difference (ZD) wide-lane filter, a single-difference (SD)wide-lane filter, or a double-difference (DD) wide-lane filter, or allof the foregoing filters to determine ZD WL ambiguities, SD WLambiguities, and/or DD WL ambiguities. The single-difference wide-laneambiguity and variance/co-variance matrix for each constellation isderived from the wide-lane filter 40, such as a zero-differenced,wide-lane float ambiguity Kalman filter.

In one embodiment, a LAMBDA (Least-squares AMBiguity DecorrelationAdjustment) or a modified LAMBDA procedure is performed to resolve theWL ambiguities. For instance, the error minimization of the leastsquares equation for decorrelated ambiguities is carried out over asearch region determined by variance and covariance matrix of theambiguities; floating ambiguity estimates and associatedvariance/co-variance matrices can be used as inputs to the LAMBDAprocess, where the output is integer ambiguity estimates.

After passing the ambiguity resolution validation (e.g., consistent withthe LAMBDA or modified LAMBDA process or an evaluation of the standarddeviation of the resolved ambiguity candidates is less than a thresholdfractional number of cycles over a minimum number of consecutiveepochs), an integer constraint representing the single differencing ofthe float wide-lane ambiguities can be applied into the float ambiguityfilter based on Equation (8). The fixed single differencing wide-laneambiguities will be used for ambiguity fixing for the reference receiver130 and the correction generation of the reference receiver 130 (e.g.,virtual base station) to be used in the correction data 108 (e.g.,distributed to any mobile receivers).

In one embodiment, the above WL filter 40 uses the WL equations to speedup convergence on the resolution of the WL ambiguities and to provideconstraints or inputs for the narrow-lane filter 44 and narrow-laneambiguity resolution, which can provide greater potential accuracy inposition estimates because the WL carrier phase measurements areassociated with more phase noise than the NL carrier phase measurements.

In one embodiment, the narrow-lane filter 44 can use the followingequations described below. The refraction corrected (RC) measurement isformed with the advantage eliminating the first order of ionospheredelay error. The RC code measurements using Equation (1-2) are formed inEquation (15) as below, which is in meter-level accuracy but unbiased.

$\begin{matrix}{P_{RC}^{j} = {{{\frac{f_{1}^{2}}{f_{1}^{2} - f_{2}^{2}}P_{1}^{j}} - {\frac{f_{2}^{2}}{f_{1}^{2} - f_{2}^{2}}P_{2}^{j}}} = {\rho^{j} + \tau_{r} + b_{RC} + \tau^{j} + B_{RC}^{j} + T + ɛ_{P_{RC}}^{j}}}} & (3)\end{matrix}$where:

-   -   b_(RC) is the receiver refraction-corrected code bias (one per        receiver and constellation for all visible CDMA satellites, such        as GPS satellites) which is a refraction-corrected (RC)        combination of the L1 receiver code bias and the L2 receiver        code bias;    -   B_(RC) ^(j) the satellite code bias which is a        refraction-corrected (RC) combination of the L1 satellite code        bias and the L2 satellite code bias;    -   ε_(P) _(RC) ^(j) is the refraction-corrected (RC) code        measurement error for satellite j including white noise,        multipath and remaining un-modeled errors; and    -   the other variables or parameters have the same meaning as set        forth earlier in this document.        In Equation 15, b_(RC) can be lumped into the receiver clock        state and can be estimated together as receiver clock nuisance        parameters. For GLONASS satellites, an additional inter-channel        code bias may be required to be estimated if the magnitude of        the inter-channel code bias is significant. B_(RC) ^(j) can be        lumped into the satellite correction τ^(j) when they are        estimated together in satellite clock determination by the PPP        network. Accordingly, for simplicity, the bias b_(RC) and B_(RC)        ^(j) in Equation (15) can be ignored and shown in Equation (16),

$\begin{matrix}{P_{RC}^{j} = {{{\frac{f_{1}^{2}}{f_{1}^{2} - f_{2}^{2}}P_{1}^{j}} - {\frac{f_{2}^{2}}{f_{1}^{2} - f_{2}^{2}}P_{2}^{j}}} = {\rho^{j} + \tau_{r} + \tau^{j} + T + ɛ_{P_{RC}}^{j}}}} & (16)\end{matrix}$where:

-   -   P_(RC) ^(j) is the refraction-corrected phase code (or        pseudo-range) for satellite j;    -   P₁ ^(j) is the measured phase code or measured pseudo-range on        the L1 frequency for satellite j;    -   P₂ ^(j) is the measured phase code or measured pseudo-range on        the L2 frequency for satellite j;    -   ε_(P) _(RC) ^(j) is the RC phase code measurement error for        satellite j including white noise, multipath and remaining        un-modeled errors; and the other variables are defined below        Equation 17.

The refraction-corrected (RC) carrier phase measurement, L_(RC) ^(j) forsatellite j, using Equations (3-4) is also formed in Equation (17) asbelow, which is in centimeter-level accurate but biased by an ambiguityterm {circumflex over (N)}_(NL) ^(j)λ_(NL).

$\begin{matrix}{L_{RC}^{j} = {{{\frac{f_{1}^{2}}{f_{1}^{2} - f_{2}^{2}}L_{1}^{j}} - {\frac{f_{2}^{2}}{f_{1}^{2} - f_{2}^{2}}L_{2}^{j}}} = {\rho^{j} + \tau_{r} + b_{NL} + \tau^{j} + B_{NL}^{j} + T + {( {N_{RC}^{j} + W^{j} + w} )\lambda_{NL}} + ɛ_{L_{RC}}^{j}}}} & (4)\end{matrix}$where:

-   -   f₁ is the L1 carrier frequency and f₂ is the L2 carrier        frequency of the received satellite signals;    -   L₁ ^(j) is the measured carrier phase for the L1 carrier        frequency transmitted from satellite j;    -   L₂ ^(j) is the measured carrier phase for the L2 carrier        frequency transmitted from satellite j;    -   ρ^(j) is the geometric distance between the satellite j phase        center and the receiver phase center including satellite        StarFire orbital correction, receiver tide displacement and        earth rotation correction;    -   τ_(r) is the receiver r clock bias or error for a given GNSS        system, where one receiver clock bias is estimated for each GNSS        system such as GPS, GLONASS, Galileo or Beidou constellation;    -   τ^(j) is the satellite clock error;    -   b_(NL) is the receiver narrow-lane phase bias (one per receiver        and constellation for all visible satellites),    -   B_(NL) ^(j) is the satellite j narrow lane phase bias (one per        satellite for all receivers), which is a RC combination of the        L1 satellite phase bias and the L2 satellite phase bias;    -   T is the tropospheric delay, and is divided into a dry component        T_(dry) and a wet component T_(wet);    -   W^(j) and w are phase windup errors for both satellite j and        receiver, in cycles, respectively, which can be corrected with        models;    -   N_(RC) ^(j) is the refraction-corrected (RC) carrier phase        ambiguity term;

$\lambda_{NL} = \frac{c}{f_{1} + f_{2}}$is the narrow lane wavelength;

-   -   and    -   ε_(L) _(RC) ^(j) is the RC phase measurement error for satellite        j including white noise, multipath and remaining un-modeled        errors.

In Equation 17, b_(NL) is a RC combination of the L1 receiver phase biasand the L2 receiver phase bias. If the b_(NL) is lumped into thefloating ambiguity state, the b_(NL) in Equation (17) can be ignored.However, this model implies that an individual ambiguity does not havean integer characteristic. Similar to the case of WL, single-differencednarrow-lane ambiguities between satellite still hold the integerproperty.

Both satellite and receiver narrow lane biases are not constant overtime. The satellite j narrow lane bias also represents the fractionalpart of the difference of code-based clock and integer phase-basedclock. If the satellite code bias B_(RC) ^(j) is combined into thesatellite clock correction, the B_(NL) ^(j) in Equation (17) will becomethe difference of B_(NL) ^(j)−B_(RC) ^(j).

N_(RC) ^(j) is the RC carrier phase ambiguity term in Equation (18), asbelow

$\begin{matrix}\begin{matrix}{{N_{RC}^{j}\lambda_{NL}} = {{\frac{f_{1}^{2}}{f_{1}^{2} - f_{2}^{2}}N_{1}^{j}\lambda_{1}} - {\frac{f_{2}^{2}}{f_{1}^{2} - f_{2}^{2}}N_{2}^{j}\lambda_{2}}}} \\{= {{\frac{N_{NL}^{j}}{2}\lambda_{WL}} + {\frac{N_{WL}^{j}}{2}\lambda_{NL}}}} \\{= {\lambda_{NL}( {N_{1}^{j} + {\frac{f_{2}}{f_{1} - f_{2}}N_{WL}^{j}}} )}} \\{= {\lambda_{NL}( {N_{2}^{j} + {\frac{f_{1}}{f_{1} - f_{2}}N_{WL}^{j}}} )}}\end{matrix} & (18)\end{matrix}$The RC carrier phase ambiguity term N_(RC) ^(j)λ_(NL) can be furtherdivided into two integer ambiguity terms. There are three equivalentcombination forms, as shown in Equation (18):

-   -   (1) Combination of integer WL ambiguity N_(WL) ^(j) in        Equation (10) and NL ambiguity N_(NL) ^(j)=N₁ ^(j)+N₂ ^(j);    -   (2) Combination of integer WL ambiguity N_(WL) ^(j) and integer        L1 carrier phase ambiguity N₁ ^(j); and    -   (3) Combination of integer WL ambiguity N_(WL) ^(j) and integer        L2 carrier phase ambiguity N₂ ^(j)

Both the WL/NL ambiguity integer N_(WL) ^(j) and N_(NL) ^(j) or L1/L2carrier phase ambiguity integer N₁ ^(j) N₂ ^(j) can be resolved toimprove position accuracy and reduce pull-in time. As long as the biasterms are removed from the refraction-corrected (RC) phase measurements,the high accuracy carrier phase measurement can be used to providecm-level positioning. The narrow lane wavelength is much shorter than WLwavelength. In the case of GPS, the narrow lane wavelength is about 10.7cm while WL wavelength is 86.4 cm. Therefore, in comparison with N_(NL)^(j) the GPS WL ambiguity integer N_(WL) ^(j) can be resolved relativelyeasier. In order to recover the integer property of the RC carrier phaseambiguity term N_(RC) ^(j), the WL ambiguity integer N_(WL) ^(j) need beresolved at first.

In one embodiment, Equations (16-17) can be used for the narrow-lanefilter 44. The zero differencing refraction-corrected code and phase rawobservations are used for the narrow-lane filter 44 measurement update.Accordingly, in the backup data the stored narrow-lane ambiguities orstored refraction-corrected ambiguities can be used to derive thenarrow-lane ambiguities. The state variables include receiver positionand velocity, receiver clock offsets, residual troposphere delay andfloating refraction-corrected ambiguities (which implicitly combineinteger wide-lane and narrow-lane ambiguities in Equation (18) alongwith receiver phase bias). For GLONASS satellites, the additionalinter-channel code bias per satellite may be required to be estimated ifthe magnitude of inter-channel code bias is significant.

The refraction-corrected phase measurements may compensate for firstorder ionospheric delay, or residual tropospheric delay that is notincluded in modeled tropospheric delay, T, or both. In one embodiment,the troposphere zenith delay and/or horizontal gradient coefficients canbe estimated after an a priori troposphere model is applied. It shouldbe noted that the receiver clock term for this method can absorb thereceiver code bias. The satellite orbit, clock and satellite narrow lanebias corrections received from correction data 108 will be applied andremaining errors are reduced to sub-centimeter level.

In an illustrative configuration, the narrow-lane filter 44 can beginprocessing even before the satellite narrow-lane bias corrections arereceived or if they are invalid. The float narrow-lane ambiguity isadjusted by the satellite narrow-lane bias when it changes state to avalid state from an invalid state. Likewise, the satellite narrow-lanebias is adjusted or removed from the float narrow-lane ambiguity whenthe narrow-lane bias changes state to an invalid state from a validstate. In one embodiment, the float ambiguity is adjusted whenever +/−2cycle jumps of satellite narrow-lane bias are detected. The adjustmentof the satellite narrow-lane bias is limited to +/−2 cycles to reducethe bandwidth or resources for data processing.

In one embodiment, a Best Integer Equivariant (BIE) or a modified BestInteger Equivariant algorithm can be used to take advantage of theinteger nature of the ambiguities to speed up the pull-in time andimprove the overall positioning accuracy.

The observation model based on Equations (16-17) allows the estimationof the position coordinate, receiver clock offset, and floatingambiguities (each combining an integer narrow lane ambiguity with thereceiver phase bias). The troposphere delay can be modeled or estimatedalong with other parameters. It should be noted that receiver clock termfor this method can absorb the receiver code bias. The satellite orbit,clock and satellite narrow lane bias correction can be obtained from thecorrection data 108 are applied and the remaining errors are reduced tosub-centimeter level.

In a summary, the ambiguities can be resolved in two steps:

-   -   (1) The first step is wide-lane ambiguity resolution using the        Equation (8). For example, details of wide-lane ambiguity        resolution are discussed in conjunction with the wide-lane        filter 40 in this document.    -   (2) The second step is the narrow-lane ambiguity resolution. For        example, details of the narrow-lane ambiguity resolution are        discussed in conjunction with the narrow-lane filter 44 in this        document. The narrow-lane ambiguities are computed efficiently        (e.g., on a constrained basis) by inserting the resolved integer        wide-lane ambiguities into equation (18). This narrow lane        ambiguity can be the integer ambiguity value associated with        either the L1 or the L2 frequency, or the narrow lane        combination of both the L1 and L2 frequencies. The effective        narrow lane ambiguity wavelength is about 10.7 cm, which is        independent of which of the narrow lane ambiguities is resolved.        This narrow lane wavelength is easily computed for either the L1        or the L2 ambiguity using equation (18). If the narrow lane        combination of both the L1 and L2 frequencies is used, the        combined narrow-lane ambiguity wavelength is only one-half of        the wavelength of an individual frequency narrow lane ambiguity.        However, since the combined narrow-lane ambiguity must have the        same odd-even integer characteristic as the wide lane ambiguity,        the same effective wavelength (10.7 cm) results for the combined        narrow-lane ambiguity and the individual narrow-lane ambiguity        because of the odd-even constraint.

In one embodiment, the reference receiver 130 that resolves wide-laneambiguities and narrow-lane ambiguities to arrive at arefraction-corrected narrow-lane ambiguity solution and associatedprecise position estimate can be used to form or generate correctiondata 108 for use by one or more rovers 22\ or mobile receivers in anetwork. At a reference receiver 130 after the narrow-lane filter 44converges (for example, the position error is less than 10 cm) fromreference receiver 130, raw measurement corrections can be generatedbased on the Equation (1-4). In an illustrative configuration, thecorrection data 108 can comprise one or more of the following: theconverged position estimate of the reference receiver 130, residualtroposphere delay, refraction-corrected ambiguities from Equation (17),fixed wide-lane ambiguities and covariance, along with raw measurementcorrections. This correction data 108 can be broadcast via wirelesscommunications devices and/or a wireless communications network 36 forother receivers nearby.

The reference station and the mobile station apply correction data 108to the carrier phase measurements, the code phase measurements, or both.The correction data 108 contains corrections for one or more of thefollowing: satellite orbit corrections, clock corrections, tidecorrections (e.g., solid Earth tide, ocean tide and polar tide), bothreceiver and satellite antenna phase center variation and offset, andboth receiver and satellite phase wind-up.

In one embodiment, the estimated parameters, such as receiver position,GNSS receiver clocks and troposphere delay are required to be corrected.The code biases for both satellite and receiver in Eq. (1-2), the phasebiases in Eq. (3-4) for both satellite and receiver and ionosphere delayare uncorrected. The integer ambiguities and receiver phase bias for thecarrier phase measurements remain in the phase corrections. Theconverged position, residual troposphere delay, refraction-correctedambiguities in Equation (17), the fixed wide-lane ambiguities inEquation (8) and their variance information are combined with the rawmeasurement corrections as backup data for the mobile receiver 20, or ascomponents in the correction data 108 to be distributed to mobilereceivers 20.

If a receiver or method of this disclosure experiences a temporary lossor interruption of one or more received satellite signals (e.g., GNSSsignals) for short time (e.g., for a few minutes) and subsequentlyregains one or more of the received satellite signal after the briefloss or interruption, the receiver or method can employ an innovativeRapid Recovery technique of recovering the converged position solution(e.g., PPP solution or position estimate) rapidly to approximately thesame level of accuracy before the signal blockage by estimating thedifferences of navigation states, such as position, carrier phaseambiguities, or the like. In accordance with one embodiment, the RapidRecovery technique has three steps or components comprising thefollowing: (1) virtual base correction generation, (2) RTK ambiguity fixand (3) Rapid recovery process.

Virtual Base Correction Generation

The backup data can include but is not limited to refraction corrected(RC), narrow-lane (NL) resolved ambiguities, in accordance with Equation17. After the narrow-lane (NL) filter converges (e.g., the positionerror is less than 10 centimeters), the mobile receiver generates backupdata (e.g., measurement corrections for the first measurement time) orupdates the backup data (e.g., at regular intervals with retention ofonly the last or most current backup data) based on the Equations (1-4).The mobile receiver 20 stores the backup data, associated with a firstmeasurement time (e.g., t₁) in a data storage device 62 for potentialuse to recover from any signal blockage events or interruptions of oneor more received satellite signals from satellite transmitters 100 atthe mobile receiver 20. The backup data can also be referred to asvirtual base station correction data. The backup data may include one ormore of the following: the converged position, residual tropospheredelay, refraction-corrected ambiguities from Equation (17), ionosphericdelay rate estimate, fixed wide-lane ambiguities and covariance, and rawmeasurement corrections.

In one embodiment, correction data 108 (as opposed to backup data)comprises satellite orbit and clock corrections that can be applied tothe raw phase measurements at the mobile receiver. In anotherembodiment, the correction data 108 comprises one or more of thefollowing: satellite orbit and clock corrections, tide corrections,receiver antenna phase center variation, receiver antenna phase centeroffset, satellite phase center variation, satellite antenna phase centeroffset, receiver phase wind-up, and satellite phase wind-up. Forexample, tide correction data 108 can comprise any of the followingitems: solid Earth tide, ocean tide and polar tide. The correction data108 is received by the correction wireless device 26, which isco-located with the mobile receiver 20, and the mobile receiver 20 canapply to the correction data 108 to the raw phase measurements of one ormore satellite signals, or both raw phase measurements and raw codephase measurements. The mobile receiver 20 applies the correction data108 in conjunction with the saved backup data to rapidly recover theprecise position of the mobile receiver, such as a precise positionbased on a fixed or resolved ambiguity in the carrier phase of thereceived satellite signals.

Certain parameters cannot be measured directly as observables at themobile receiver 20 or any reference receiver 130. The mobile receiver 20estimates estimated parameters, such as receiver position, (GNSS)receiver clock, and troposphere delay from the observables, or theobservables in combination with models. After considering the correctiondata 108 and the estimated parameters, the mobile receiver 20 hascertain biases. For example, the code biases for both satellite andreceiver in Equations (1-2) are not corrected, the phase biases inEquations (3-4) for both satellite and receiver and ionosphere delay arenot corrected. The correction data 108 does not contain integerambiguities and receiver phase bias for the carrier phase measurements.The integer ambiguities and receiver phase bias remain in the correctiondata 108 that is applied at the mobile receiver 20.

Ionospheric Estimation/Modeling

In one embodiment, at the mobile receiver 20 in the relative positioningmodule 18, the navigation positioning estimator 50, or the atmosphericbias estimator 42 estimates the atmospheric delay or bias between timet₁ and t₂, where t₁ represent the time prior to signal loss orinterruption and t₂ represent the time after the signal loss orinterruption. The atmospheric dependent errors such as ionosphere error(i.e. I) should be handled independently at each time t₁ and t₂. In oneembodiment, the first strategy generally used to mitigate this portionof errors is a combination of modeling and estimation. The ionospheredelay error increases over time quickly (e.g., roughly speaking 10centimeters per minute). When dual frequency carrier phase measurementsfrom Equations (3-4) are available from a receiver (e.g., mobilereceiver), the difference of geometry free phase measurements betweensatellites i and j over t₁ and t₂ can be expressed as Equation (19) atfollows:

$\begin{matrix}{{{{\nabla\Phi_{1}^{ij}}{\lambda_{1}( t_{2} )}} - {{\nabla\Phi_{2}^{ij}}{\lambda_{2}( t_{2} )}} - {{\nabla\Phi_{1}^{ij}}{\lambda_{1}( t_{1} )}} + {{\nabla\Phi_{2}^{ij}}{\lambda_{2}( t_{1} )}}} = {{\frac{f_{1}^{2} - f_{2}^{2}}{f_{2}^{2}}{\nabla\Delta}\; I_{t_{1},t_{2}}^{ij}} + {\nabla{B_{L_{1}}^{ij}( t_{2} )}} - {\nabla{B_{L_{1}}^{ij}( t_{1} )}} - {\nabla{B_{L_{2}}^{ij}( t_{2} )}} + {\nabla{B_{L_{2}}^{ij}( t_{1} )}} + {\nabla{W_{t_{2}}^{ij}( {\lambda_{1} - \lambda_{2}} )}} - {\nabla{W_{t_{1}}^{ij}( {\lambda_{1} - \lambda_{2}} )}}}} & (19)\end{matrix}$Where ∇ is difference operator between satellites i and j. ∇ΔI_(t) ₁_(,t) ₂ ^(ij) is ionosphere change (e.g., double-difference ionosphereestimate) between satellites i and j over t₁ (before shading or signalinterruption) and t₂ (after shading or signal interruption). Equation 19has a first difference in phase wind-up W_(t1) ^(ij) between satellitesi and j over t₁ (before shading or signal interruption) and t₂ (aftershading or signal interruption), and second difference in phase wind-upW_(t) ₂ ^(ij) between satellites i and j over t₁ (before shading orsignal interruption) and t₂ (after shading or signal interruption).

At the mobile receiver 20, the receiver phase wind-up and receiver phasebias can be cancelled between the satellites (100). The satellite phasebias and phase wind-up change very slowly over time. One of majorfactors causing the phase wind-up change is attributable to satelliteyaw rotation, which is less than the maximum yaw rate 0.2 degree persecond. For simplicity, the satellite phase bias and phase windupchanges can thus be ignored. The above-mentioned equation (19) can thenbe simplified as Equation (20)

$\begin{matrix}{{{{\nabla\Phi_{1}^{ij}}{\lambda_{1}( t_{2} )}} - {{\nabla\Phi_{2}^{ij}}{\lambda_{2}( t_{2} )}} - {{\nabla\Phi_{1}^{ij}}{\lambda_{1}( t_{1} )}} + {{\nabla\Phi_{2}^{ij}}\lambda_{2}( t_{1} )}} = {\frac{f_{1}^{2} - f_{2}^{2}}{f_{2}^{2}}{\nabla\Delta}\; I_{t_{1},t_{2}}^{ij}}} & (20)\end{matrix}$The ionosphere change ∇ΔI_(t) ₁ _(,t) ₂ ^(ij) can be modeled as afirst-order Gauss-Markov process as Equation (21) as follows:∇I ^(ij)(t ₂)=ϕ_(t) ₁ _(,t) ₂ ·∇İ ^(ij)(t ₁)+Q _(t) ₁ _(,t) ₂ , whereϕ_(t,t−1) ≅e ^(−τ(t) ² ^(−t) ¹ ⁾ ,Q _(t) ₁ _(,t) ₂ =q _(I) ²(t ₂ −t₁)  (21)where τ, σ are correlation time and dynamics of the ionosphere ratewhich are design factors characterizing the model, and Q_(t) ₁ _(,t) ₂is a time-varying, first-order Gauss-Markov, function based on theionosphere parameter q_(I) ². In the navigation positioning estimator 50or the atmospheric bias estimator 42, the measurements from Equation(20) can be used for measurement update and Equation (21) can be usedfor time update, respectively, of the precise positioning module 16, theatmospheric bias estimator 42, or a predictive filter (e.g., Kalmanfilter) used to estimate this ionospheric delay change. Note that onlyincremental or relative ionospheric values (e.g., ΔI=I(t₂)−I(t₁)) areinteresting, but not the absolute ionospheric error itself (e.g., I(t₂)or I(t₁)). It should be emphasized that in order to model the ionosphererate properly, the differencing operation between satellites is requiredin order to eliminate the receiver phase bias and receiver phase wind-upchanges over time.

In the mobile receiver 20, navigation positioning estimator 50 orprecise positioning module 16 determines or estimates the backup datafor the first measurement time (t₁). For example, for the firstmeasurement time, t₁ (e.g., prior to signal loss or interruption), themobile receiver 20 determines and the backup/recovery module 46 stores,retrieves or manages the backup data, such as converged position of themobile receiver 20 residual troposphere delay, refraction-correctedambiguities in the equation (17), the ionosphere rate in Equation (21),the fixed wide-lane ambiguities in the Equation (8) and their varianceinformation are combined with the raw measurement corrections. Thebackup data for the first measurement time (e.g., t₁) is stored in thedata storage device 62 of the mobile receiver 20, which can be modeledas a virtual base station or virtual reference receiver for time t₁.

In addition to the signal blockage, the occurrence of a power outage canalso cause the precise positioning module 16 (e.g., a precise pointposition (PPP) navigation module or estimator) reset and correspondinglong convergence time. For example, a power outage of the mobilereceiver can occur because of power-connectivity issue (e.g., brokenwire in a wiring harness or oxidized connector), battery discharge,circuit breaker, blown fuse, main power cut-off from a vehicle (e.g.,agricultural work vehicle) to the mobile receiver. Accordingly, incertain embodiments, the data storage device 62 may comprisenon-volatile random memory or flash memory such that the backup data(e.g., virtual base correction) for Rapid Recovery (RR) before shadingcan be read, retrieved or recovered after the mobile receiver is powercycled within a short gap, such as approximately three (3) toapproximately five (5) minutes. As a power loss can occur anytime, thevirtual base corrections are written to data storage device (e.g.,NVRAM) in a regular time interval based on user settings or factorydefault settings, such as approximately every 30 seconds. Further, in onembodiment, the previously recorded backup data for the firstmeasurement time t₁ is refreshed or overwritten with new or currentbackup data for the measurement time for each regular time interval, orfor a separate overwrite time interval that exceeds the regulator timeinterval.

RTK Ambiguity Resolution

In one embodiment, the relative positioning module 18 comprises areal-time kinematic (RTK) filter or relative position estimator. Ingeneral, the relative positioning module 18 comprises a predictivefilter, such as Kalman filter or a modified Kalman filter. RTK algorithmis a reliable method for determining the relative position and ambiguitydifference between two receivers in carrier-phase positioning inreal-time (e.g., for navigation applications). If the reception of thereceived satellite signals is interrupted at the mobile receiver,because of a temporary power loss of input power to the mobile receiveror obstruction of the received satellite signal from shading (e.g.,trees or vegetation), there will be a measurement gap (e.g., GNSSmeasurement gap) in the raw phase measurements and raw code phasemeasurements at the mobile receiver. Further, the ambiguities beforeshading at the first measurement time and the ambiguities after shadingat the second measurement time may change. Therefore, for precisepositioning the mobile receiver 20, the precise positioning module 16 orthe relative positioning module 18 determines new ambiguities and thecycle slip detector 59 detects cycle slips. However, some error sourcessuch as the satellite-dependent code and phase biases before shading donot change significantly over a short time from when the satellite orbitand clock corrections are applied; the mobile receiver mitigates theeffects of changes over time (e.g., between the first measurement timeand the second measurement time) by applying a time differencingoperator between epochs (e.g., between the first measurement time andthe second measurement time). In order to handle this situationeffectively, the mobile receiver 20 or the relative positioning module18 uses an RTK technique or filter. The purpose of the RTK ambiguityresolution is to compute the relative position change from before andafter shading ΔX_(RTK), as well as the double-difference (DD)ambiguities for wide-lane (WL) and refraction-corrected (RC), ∇Δ_(WL)^(ij) (RTK) and ∇Δ_(RC) ^(ij) (RTK). Here, the RTK is not used todetermine the relative position and relative position and ambiguitydifference between two receivers in carrier-phase positioning. Instead,the backup data (e.g., previous measurement corrections) from the firstmeasurement time are is considered as a virtual base station. Actually,the double differencing approach between epochs (t₁ and t₂) and betweensatellites, similar to RTK, are used.

The RTK algorithm or RTK-like algorithm uses the followingdouble-difference equations for code phase and carrier phase todetermine the relative position vector (e.g., baseline vector) betweenthe reference receiver 130

$\begin{matrix}{{{\nabla\Delta}\; P_{1}^{ij}} = {{\nabla{\Delta\rho}^{ij}} + {{\nabla\Delta}\; I^{ij}} + {\nabla{\Delta ɛ}_{P_{1}}^{ij}}}} & (22) \\{{{\nabla\Delta}\; P_{2}^{1}} = {{\nabla{\Delta\rho}^{j}} + {\frac{f_{1}^{2}}{f_{2}^{2}}{\nabla\Delta}\; I^{ij}} + {\nabla{\Delta ɛ}_{P_{2}}^{j}}}} & (23) \\{{{\nabla\Delta}\; L_{1}^{ij}} = {{\nabla{\Delta\rho}^{ij}} - {{\nabla\Delta}\; I^{ij}} + {{\nabla N_{1}^{i}}\lambda_{1}^{i}} - {\Delta\; N_{1}^{j}\lambda_{1}^{j}} + {\nabla{\Delta ɛ}_{L_{1}}^{ij}}}} & (24) \\{{{\nabla\Delta}\; L_{2}^{ij}} = {{\nabla{\Delta\rho}^{ij}} - {\frac{f_{1}^{2}}{f_{2}^{2}}{\nabla\Delta}\; I^{ij}} + {\Delta\; N_{2}^{i}\lambda_{2}^{i}} - {\Delta\; N_{2}^{j}\lambda_{2}^{j}} + {\nabla{\Delta ɛ}_{L_{2}}^{ij}}}} & (25)\end{matrix}$

-   -   ∇ΔL₁ ^(ij) is the double-difference L1 carrier phase        measurements with respect to satellite i and j and the mobile        receiver at the measurement times t₁ and t₂, where t₁ is the        first measurement time and t₂ is the second measurement time;    -   ∇Δρ^(ij) is the double-difference geometric distance between the        satellite j phase center and the mobile receiver phase center        and between satellite i phase center and the receiver phase        center, including satellite orbital correction, receiver tide        displacement and earth rotation correction;    -   ∇ΔI^(ij) is the double-difference ionosphere error for a given        satellite j and satellite i;    -   ∇N₁ ^(i)λ₁ ^(i) is the single-differenced integer ambiguity for        satellite i multiplied by the wavelength for the L1 carrier from        satellite i;    -   ΔN₁ ^(j)λ₁ ^(j) is the single-differenced integer ambiguity for        satellite j multiplied by the wavelength for the L1 carrier from        satellite j;    -   ∇Δε_(L) ₁ ^(ij) is the double-difference phase measurement error        for satellite j and satellite i including white noise, multipath        and remaining un-modeled errors with respect to the L1        frequency;    -   ∇ΔL₂ ^(ij) is the double-difference L2 carrier phase        measurements with respect to satellite i and j, and the mobile        receiver at the measurement times t₁ and t₂ where t₁ is the        first measurement time and t₂ is the second measurement time;    -   ΔN₂ ^(i)λ₂ ^(j) is the single-differenced integer ambiguity for        satellite i multiplied by the wavelength for the L2 carrier from        satellite i;    -   ΔN₂ ^(j)λ₂ ^(j) is the single-differenced integer ambiguity for        satellite j multiplied by the wavelength for the L2 carrier from        satellite j;    -   ∇Δε_(L) ₂ ^(ij) is the double-difference phase measurement error        for satellite j and satellite i including white noise, multipath        and remaining un-modeled errors with respect to the L2        frequency; and f₁ is the L1 carrier frequency and f₂ is the L2        carrier frequency of the received satellite signals.

In Equations (22-25), the receiver/satellite dependent errors, such ascode phase bias (e.g., receiver code phase bias and satellite code phasebias), carrier phase bias (e.g., receiver phase bias and satellite phasebias) and clock bias (e.g., receiver clock bias and satellite clockbias), that are common between satellites and the mobile receiver atmeasurement times t₁ and t₂ (e.g., measurement times or epochs) can becancelled out by the double differencing operation between satellitesand receiver at measurement times. t₁ and t₂ where t₁ is the firstmeasurement time and t₂ is the second measurement time.

Under certain conditions, the ionosphere error can be significant;mobile receiver 20 can compensate for the ionosphere error based on thetime difference and the ionosphere rate estimated in the Equation (21).The remaining ionosphere error is also estimated per satellite in theRTK engine. The troposphere error is corrected based on the elevationmapping function using the troposphere bias estimation before theshading events or power cycle. In one embodiment, the mobile receiver 20can ignore the remaining satellite dependent errors including orbit,clock, code bias and phase biases after the time differencing operatoris conducted and orbit and clock corrections from the correction data108 are applied at both epochs.

Although two GNSS receivers (e.g., reference receiver 130 and rover 20)and two satellites are usually required for the formation of adouble-difference measurements, here one GNSS receiver (e.g., mobilereceiver 20 or rover) takes measurements at two different measurementtimes for two different satellites to form double-differencemeasurements. In one embodiment in accordance with the RTK algorithm, aminimum of four double-difference equations and five satellites arerequired to solve for the relative position vector and the associatedinteger ambiguity for a three dimensional position estimates (e.g., inCartesian coordinates, x, y, z).

The correction data 108, which includes orbit and clock corrections, areapplied for the mobile receiver between the two measurement times orepochs (e.g., between the first measurement time and the secondmeasurement time). It should be mentioned that the different GLONASSsatellites have different frequency and wavelength. Accordingly, forGLONASS satellites, the receiver clock error after thedouble-differencing phase measurements can be cancelled, but theresulting double differencing ambiguities are not integers any more. Thefloat ambiguity bias for each reference satellite (e.g., GLONASSsatellite) is required to be estimated.

The relative positioning module 18 or an RTK algorithm can be used toestimate the difference of integer ambiguity or cycle slips between t₁and t₂, remaining ionosphere delay bias per satellite, relative positionchange (of the mobile receiver 20) ΔX_(RTK)=X(t₂)−X(t₁) at t₂ from theposition at time t₁. X(t₁) is a known position, or rather a convergedposition, with precise (e.g., PPP) centimeter level accuracy if theprecise positioning module (e.g., PPP module) or the predictive filterfor Equation (17) is already converged.

In one embodiment, the standard LAMBDA method, the least squares method,or another ambiguity resolution technique can be applied to resolve theDD ambiguities. If the ambiguity resolution succeeds, the relativeposition accuracy of ΔX_(RTK) can be determined to centimeter level aswell. Therefore, precise position X(t₂) of the mobile receiver or roverat the second measurement time can be derived at centimeter levelaccuracy. If the DD L1 ambiguities and DD L2 ambiguities (e.g., orambiguity changes) can be fixed correctly in Equations (24-25), thedouble difference of wide lane integer ambiguity and refractioncorrected ambiguity between satellite i and j and between measurementtimes t₁ and t₂ can be derived using Equations (26-27):

$\begin{matrix}{{{\nabla\Delta}\;{N_{WL}^{ij}({RTK})}} = {{{\nabla\Delta}\; N_{1}^{ij}} - {{\nabla\Delta}\; N_{2}^{ij}}}} & (26) \\{{{\nabla\Delta}\;{N_{RC}^{ij}({RTK})}} = {{\frac{f_{1}^{2}}{f_{1}^{2} - f_{2}^{2}}{\nabla\Delta}\; N_{1}^{ij}} - {\frac{f_{2}^{2}}{f_{1}^{2} - f_{2}^{2}}{\nabla\Delta}\; N_{2}^{ij}}}} & (27)\end{matrix}$

-   -   where:    -   ∇ΔN_(WL) ^(ij)(RTK) is the RTK double-difference wide-lane        ambiguity for satellites i and j with respect to the mobile        receiver at measurement times t₁ and t₂ where t₁ is the first        measurement time and t₂ is the second measurement time;    -   ∇ΔN₁ ^(ij) is the double-difference L1 wide-lane ambiguity for        satellites i and j with respect to the mobile receiver        measurement times t₁ and t₂ where t₁ is the first measurement        time and t₂ is the second measurement time;    -   ∇ΔN₂ ^(ij) is the double-difference L2 wide-lane ambiguity for        satellites i and j with respect to the mobile receiver a        measurement times t₁ and t₂ where t₁ is the first measurement        time and t₂ is the second measurement time;    -   f₁ is the frequency of the L1 carrier phase signal and f₂ is the        frequency of the L2 carrier phase signal; and    -   ∇ΔN_(RC) ^(ij)(RTK) is the RTK refraction-corrected        double-difference float ambiguity (e.g., wide-lane or        narrow-lane) for satellites i and j with respect to the mobile        receiver at measurement times t₁ and t₂.

Recovery Process

After RTK ambiguity resolution is successfully completed, the followingconstraint process called rapid recovery will start. In one embodiment,the rapid recovery process comprises the following steps: (1) singledifference wide-lane ambiguity recovery, (2) current mobile receiverposition or rover position recovery (e.g. at the second measurementtime, t₂), (3) troposphere bias constraints, (4) single differencerefraction-corrected ambiguity constraints and (5) constraint outlierdetection and adaptation. The high-level purpose of the rapid recoveryprocess is to allow for seamless precise positioning module (e.g., PPPestimator) recovery based on the RTK results from Equations 24-27.

(1) Single-Difference Wide-Lane Ambiguity Recovery

The resolution of single-differenced ambiguity between satellites arepreferred over the zero-difference ambiguity because the zero-differenceambiguity contains both integer ambiguity value and a common receiverphase bias or error, which can change rapidly.The SD wide-lane ambiguity ∇N_(WK) ^(ij)(t₁) can be fixed into integervalue in Wide-lane filter using the Eq. (8) before the shading event orat the first measurement time, t₁. The fixed DD wide-lane ambiguity∇ΔN_(WL) ^(ij)(RTK) in Eq. (26) and ∇N_(WL) ^(ij)(t₁) can be used torecover current wide-lane SD integer ambiguity. The SD wide-lane integerambiguity can be computed in Eq. (28) below. The integer constraint∇N_(WL) ^(ij)(t₂) associated with the second measurement time (e.g., t₂)or ∇N_(WL) ^(ij)(PPP), associated with correction data 108 (e.g.,precise point positioning correction data 108, which remains valid fromthe second measurement time until a cycle slip) can then be applied intocurrent wide-lane filter in Equation 8.∇N _(WL) ^(ij)(PPP)=∇ΔN _(WL) ^(ij)(RTK)+∇N _(WL) ^(ij)(t ₁)or∇N _(WL) ^(ij)(t ₂)=∇ΔN _(WL) ^(ij)(RTK)+∇N _(WL) ^(ij)(t ₁)  (28)(2) Current Mobile Receiver PositionThe current precise (e.g., PPP) position X(t₂) at the second measurementtime can be derived in Equation (29) from the relative position changeΔX_(RTK)=X(t₂)−X(t₁) in the RTK solution, and RTK virtual base positionX(t₁) at time t₁ from previously converged precise (e.g., PPP) solution.X(t ₂)=ΔX _(RTK) +X(t ₁)  (29)where:X(t₂) is the position (second position or current position) of themobile receiver at the second measurement time, t₂;ΔX_(RTK) is a relative change in position of the mobile receiver that isobserved by the RTK filter between the first measurement time, t₁, andthe second measurement time, t₂; andX(t₁) is the estimated position (first position) of the mobile receiver(also referred to as virtual base) at the first measurement time,

The covariance matrix Q_(XYZ)(t₂) for current position can be derivedfrom the variance of virtual base position Q_(XYZ)(t₁) and relativeposition change Q_(ΔXYZ(RTK)) in RTK from time t₁ to t₂ respectively ifthey are assumed to be un-correlated as shown in Equation (30)Q _(XYZ)(t ₂)=Q _(ΔXYZ(RTK)) +Q _(XYZ)(t ₁)  (30)where:

-   -   Q_(XYZ)(t₂) is the covariance matrix for the estimated position        of the mobile receiver at the second measurement time;    -   Q_(ΔXYZ(RTK)) is the change in the covariance matrix associated        with the relative change in position ΔX_(RTK); and    -   Q_(XYZ)(t₁) is the covariance matrix for the estimated position        of the mobile receiver at the first measurement time.    -   The precise position X_(PPP)(t₂) (e.g., precise point position)        and corresponding covariance matrix Q_(XYZ)(t₂) can be        considered as virtual measurements or constraints to be applied        to the current narrow-lane filter in the navigation positioning        estimator 50 or in the relative positioning module 18.        (3) Tropospheric Delay Error

In certain embodiments, the remaining troposphere delay error (e.g.,additional residual tropospheric delay), after a prior modeling oftropospheric delay, T, pursuant to the equations referenced in thisdocument, can be estimated as it changes slowly over time and withtraveled distance. For example, the estimated troposphere remainsgenerally unchanged over a short time period (e.g., a few minutesbetween observed measurement times t₁ and t₂) in accordance withEquation (31). However, over a greater time period (e.g., greater than afew minutes) the variance of estimated tropospheric delay

(e.q., Q_(T_(t₂)))at subsequent measurement time (e.g., t₂) is required to be inflated byboth spatial q_(Trop) ^(Spatial) (such as 0.1 ppm) and temporalcorrelation factors q_(Trop) ^(Temporal) (such as 1 centimeter per hour)in Equation (32). The Equations (31-32) below can be considered as avirtual measurement to constrain the troposphere delay estimation in theprecise positioning estimator 16 (e.g., PPP estimator).T(t ₂)≈T(t ₁)  (31)where:

-   -   T(t₂) is the estimated tropospheric delay for the mobile        receiver at the second measurement time, t₂; and    -   T(t₁) is the estimated tropospheric delay for the mobile        receiver (e.g., by the atmospheric bias estimator 42) at the        first measurement time, t₁.

$\begin{matrix}{Q_{T_{t_{2}}} = {Q_{T_{t_{1}}} + {{\Delta\; t} \star q_{Trop}^{Temporal}} + {{{\Delta\; X_{RTK}}} \star q_{Trop}^{Spatial}}}} & (32)\end{matrix}$

-   -   where:

Q_(T_(t₂))is the variance of estimated tropospheric delay at the secondmeasurement time, t2;

-   -   q_(Trop) ^(Spatial) is a spatial correlation factor for        inflating the covariance over spatial displacement of the mobile        receiver;    -   q_(Trop) ^(Temporal) is a temporal correlation factor for        inflating the covariance over time;    -   |ΔX_(RTK)| is traveled distance of the mobile receiver from the        first measurement time, t₁, to the second measurement time, t₂,        and    -   Δt=t₂−t₁ or the time difference between the first measurement        time and the second measurement time.        (4) Single Difference Refraction-Corrected Ambiguity Constraints

In one embodiment, the SD refraction-corrected ambiguity and variance attime t₂ can be derived as shown below in Equations (33-34) based on theprevious (before signal blockage) converged RC ambiguity ∇N_(RC)^(ij)(t₁) at t₁ and DD ambiguity ∇ΔN_(RC) ^(ij)(RTK). The ∇N_(RC)^(ij)(t₁) and variance Q_(∇N) _(RC) _(ij) (t₁) can be computed as partof the virtual base correction from the narrow-lane filter as shown inEquations (16-17).∇N _(RC) ^(ij)(t ₂)=∇ΔN _(RC) ^(ij)(RTK)+∇N _(RC) ^(ij)(t ₁)  (33)Q _(∇N) _(RC) _(ij) (t ₂)=Q _(∇ΔN) _(RC) _(ij) _((RTK)) +Q _(∇N) _(RC)_(ij) (t ₁)  (34)(5) Constraint Outlier Detection and Adaptation

The Variance of ∇ΔN_(RC) ^(ij) (RTK) is zero if both L1 and L2 ambiguityare fixed into integer values. Otherwise, the variance of float ∇ΔN_(RC)^(ij) (RTK) ambiguity can derived from RTK ambiguity state variance. Itshould be mentioned that the constraint of float ambiguity from RTK canbe applied as well.

After the constraints of single-difference (SD) wide-lane (WL)ambiguities are determined, current position, troposphere andsingle-difference ambiguities are estimated by the wide lane 40 andnarrow-lane filter 44; the post-fit residuals of these constraints(e.g., SD WL ambiguities, other SD ambiguities, or DD ambiguities,respectively) can be computed by the navigation positioning estimator50. The ratio of post-fit residual, or its standard deviation, dividedby the square root of the variance (e.g., standard deviation) of theconstraint can be computed. If that ratio exceeds a large threshold suchas 3, the constraint (e.g., SD WL ambiguities, other SD ambiguities, orDD ambiguities, respectively) should be considered problematic. Theratio exceeding the threshold could be caused by incorrect RTK ambiguityresolution output of the RTK filter 48 or of the navigation positioningestimator 50, for example. The remedy for this issue of ratio exceedingthe threshold or erroneous ambiguity resolution is to remove thecorresponding problematic constraints one-by-one. An alternativeapproach is to de-weight those problematic constraint(s) by increasingtheir variance, using the ratio as a scaling factor, for example.Typically, the adaptation is done in an iterative way.

In one configuration, the recovered data includes one or more of thefollowing: (1) single difference wide-lane ambiguities, (2) currentmobile receiver position (e.g. at the second measurement time, t₂), (3)troposphere bias constraints, (4) single difference refraction-correctedambiguities and (5) variance in SD WL ambiguities, mobile receiverposition, tropospheric bias, or SD RC ambiguities. The recovered data isbased recovery data such as one or more of the following: RTK DDwide-lane integer ambiguity, ∇ΔN_(WL) ^(ij) (RTK); RTK fixedrefraction-corrected (RC) float ambiguity, ∇ΔN_(RC) ^(ij) (RTK); and therelative position ΔX (e.g., relative position vector between the mobilereceiver at the first measurement time and the second measurement time);and the variance/co-variance of the relative position ΔX. The recoverydata can be used as additional constraints or inputs to speed up currentfilter convergence (e.g., wide-lane filter convergence, narrow-lanefilter convergence, or both) process at rover in the precise positioningmodule.

FIG. 3 illustrates one embodiment of method and satellite receiver forrapid recovery of precise position by backup data upon temporary loss orinterruption of one or more received satellite signals. The method ofFIG. 3 begins in step S300. In step S300, the mobile receiver 20 (e.g.,rover) or the measurement module 56 measures carrier phase and codephase of received satellite signals at a first measurement time (e.g.,first epoch). The first measurement time is prior to signal loss,corruption, degradation, disruption, interference with, or interruptionof any of the following: (a) received satellite signals from one or moresatellite transmitters 100, (b) the correction signal from a correctiondata source (24 or 124), or (c) prior to power outage of electricalpower (e.g., direct current voltage from a vehicle) supplied to themobile receiver 20. For example, if the correction signal that carriesthe correction data 108 is from a communications satellite 135, thecorrection wireless device 26 (e.g., L-band satellite receiver) canexperience signal loss (e.g., fading or shading) simultaneously with themobile receiver 20 that receives the L1 and L2 carrier signals,depending upon the orientation and elevation of the respectivecommunications satellite 135 and applicable GNSS satellite transmitters100. In one embodiment, the carrier phase measurement module 58 measuresor attempts to measure carrier phase of the received satellite signalswithin view or reception range over a series of measurement times (e.g.,successive epochs including the first epoch or first measurement time).Further, the code phase measurement module 60 may measure or attempt tomeasure code phase or pseudo range of the received satellite signalswithin view or reception range of the series of measurement times (e.g.,successive epochs including the first epoch).

In step S302 at the mobile receiver 20 for the first measurement time,the precise positioning module 16 or navigation positioning estimator 50resolves: (1) wide-lane ambiguities and (2) narrow-lane ambiguities orrefraction-corrected ambiguities based on the measured carrier phase andcode phase of the received satellite signals, tropospheric delay models,and correction data 108 (e.g. in accordance with a precise pointpositioning algorithm). For example, the mobile receiver 20 can estimatewide-lane (WL) ambiguities (e.g., single-difference (SD) WLambiguities); narrow-lane ambiguities (e.g., single difference (SD) NLambiguities) or refraction-corrected (RC) ambiguities; a referenceposition and residual tropospheric bias at the mobile receiver 20 basedon the correction data 108 (e.g., precise point position (PPP)correction data). In one embodiment, the precise point position (PPP)correction data comprises precise clock and orbit data for respectivesatellites, such as clock data that has less than a certain errormeasured in time or range error and orbit data that meets certainstandard deviation metrics to support reliable position estimates withsub-decimeter level accuracy on a global basis. In one example,consistent with the resolved ambiguities, the precise positioning module16 or navigation positioning estimator 50 estimates the referenceposition (e.g., three dimensional coordinates), which comprises aprecise point positioning reference position of the mobile receiver 20that is stationary or mobile.

In step S304 at the mobile receiver 20 at regular intervals, the mobilereceiver 20 stores backup data in a data storage device 62 (e.g.,nonvolatile random access memory), where the backup data comprises theresolved wide-lane ambiguities; resolved narrow-lane ambiguities orresolved refraction-corrected ambiguities; estimated tropospheric delaybias, raw measured carrier phase and code phase of the receivedsatellite signals, and reference receiver 130 position. For example, thebackup data from step S302 is stored in the data storage device 62 at aregular interval (e.g., periodic basis or refresh rate, such asapproximately once every twenty seconds to every five minutes).

In step S305 at a second measurement time (e.g., second epoch after thefirst epoch) within a maximum recovery time period after the firstmeasurement time or after a detection time (e.g., first detection time)of signal loss, interruption, disruption, or corruption that immediatelyfollows the first measurement time; the mobile receiver 20 measurescarrier phase and code phase of the received satellite signals. In oneexample, the maximum recovery time period is as soon as possible withinthe maximum recovery time after return of an interrupted signal, lostsignal, or power from the power outage.

In step S306, for the second measurement time, the mobile receiver 20retrieves or reads the backup data and applies the backup data to thereal-time kinematic (RTK) algorithm or the RTK filter 48 of the relativepositioning module 18 to provide: (1) relative position vector betweenthe mobile receiver 20 at the first measurement time and the mobilereceiver 20 at the second measurement time, and (2) recovery dataassociated with the a time-differenced, satellite-differenced,double-difference measurements, for the mobile receiver, between thefirst measurement time and the second measurement time (e.g.,time-differenced) and between at least two satellites (e.g.,satellite-differenced). The recovery data is distinct from the backupdata, where the relative positioning module 18 derives from the recoverydata from the backup data and the RTK filter 48. As indicated above, therecovery data includes one or more of the following: RTKdouble-difference (DD) wide-lane integer ambiguity, ∇ΔN_(WL) ^(ij)(RTK); RTK fixed refraction-corrected (RC) double-difference (DD) floatambiguity, ∇ΔN_(RC) ^(ij) (RTK); and the relative position ΔX (e.g.,relative position vector of the mobile receiver between the firstmeasurement time and the second measurement time); and thevariance/co-variance of the relative position ΔX.

Step S306 may be carried out by various techniques, which may be appliedseparately or cumulatively. Under a first technique, the relativepositioning module 18 uses double-difference of phase measurements atthe mobile receiver 20 between the first measurement time and the secondmeasurement time and two satellites to resolve double-difference RTKambiguities.

Under a second technique, relative positioning module 18 or thereal-time kinematic filter 48 determines a relative position or therelative position vector for a mobile receiver 20 between the firstmeasurement time and the second measurement time based on a set ofreal-time kinematic (RTK) algorithms to resolve the L1/L2double-differenced (DD) fixed integer values (N₁, N₂) for the mobilereceiver 20 on the L1 frequency and reference satellite per GlobalNavigation Satellite System (GNSS) system and between the mobilereceiver 20 on the L2 frequency and the same reference satellite perGNSS system.

Under a third technique, the data storage device 62 of the rover 20stores or retrieves backup data provided from the data storage device 62to the navigation positioning estimator 50, where the backup datacomprises a set of one or more of the following: resolved wide-laneambiguities, resolved narrow-lane ambiguities, estimated troposphericdelay bias, raw measured carrier phase of the received satellitesignals, and mobile receiver position (e.g., fully or substantiallyconverged precise point position (PPP) of the mobile receiver 20).

Under a fourth example, S306, the mobile receiver 20 can receive a datamessage of backup data from a data storage device 62, where the backupdata comprises the estimated (e.g., pulled-in or post-convergence)wide-lane (WL) and narrow-lane (NL) ambiguities from one or moresatellite carrier signals, estimated (e.g., pulled-in or postconvergence) reference position or coordinates of the mobile receivernear, the tropospheric delay at zenith direction including the a priorimodel and residual tropospheric delay estimation, and raw phasemeasurements of the mobile receiver 20 at a first measurement time. Themobile receiver 20, or its relative positioning module 18, is adapted toestimate a relative position or relative position vector between amobile receiver 20 at a first measurement time and a second measurementtime based on a set of real-time kinematic (RTK) algorithm to resolvethe L1/L2 double-differenced (DD) fixed integer values (N₁, N₂) betweena reference mobile receiver at carrier frequency L1 and referencesatellite per each GNSS system (e.g., GPS or GLONASS) and between areference receiver 130 at carrier frequency L2 and the same referencesatellite per each GNSS system. The double differencing can be used toeliminate receiver clock bias and estimate frequency dependent bias inthe carrier phase measurements. The carrier phase measurements atdifferent carrier frequencies (e.g., L1, L2) can be used to estimate orcompensate for ionospheric delay.

In step S308 at the mobile receiver 20, the precise positioning module16 or the backup recovery module 46 applies the relative positionvector, the backup data, recovery data, and correction data 108 (e.g.,with precise clock and orbit information on the received satellitesignals) as inputs, constraints, or both for convergence of one or morepredictive filters on wide-lane and narrow-lane ambiguities (e.g., inaccordance with a precise positioning algorithm). For example, the rover20 or its data processor 66, is adapted to compute single-difference(SD) wide-lane ambiguity (e.g., integer ambiguity) based on recoverydata, such as a L1/L2 fixed, double-difference (DD) ambiguitiesassociated with an RTK solution (e.g., resolved RTK ambiguity in stepS306), and backup data, such as the estimated wide-lane (WL) ambiguities(e.g., floating WL ambiguity) from the data storage device 62, and therefraction-corrected (RC) ambiguities from the data storage device 62.The backup data may comprise any of the following data: the resolvedwide-lane (WL) ambiguities, resolved narrow-lane (NL) ambiguities, andthe refraction-corrected (RC) ambiguities, and raw phase measurements ofthe mobile receiver 20. If the backup data includes the resolved NLambiguities, the related refraction-corrected (RC) ambiguities for thesame satellites and measurement times are not required, and vice versa.

After RTK ambiguity resolution is successfully completed for the mobilereceiver 20 for at least five GNSS satellites, the mobile receiver 20can use the recovery data, backup data, and correction data 108 to applyrapid recovery for temporary interruption or loss of one or morereceived satellite signals. The rapid recovery supports quick, sometimesalmost immediate resumption of full ambiguity resolution or preciseposition estimation of one or more filters (e.g., 38, 40, 44) of theprecise positioning module 16. Although only an optional zero-differencefilter 38 is shown, the precise positioning module 16 may comprise oneor more SD filters, DD filters, or both to support the rapid recovery ofthe precise position or solution of the mobile receiver 20. In certainembodiments, the rapid convergence may be referred to as Rapid Recovery.

After the constraints of single-differencing wide lane ambiguities, therover 20 position, tropospheric delay bias and single-differencingnarrow-lane ambiguities are applied into the wide-lane filter 40 andnarrow-lane filter 44, the post-fit residuals of these constraints (SDWL ambiguities and SD NL ambiguities) can be computed. The ratio of thepost-fit residual, or its standard deviation (of SD WL ambiguities or SDNL ambiguities, respectively) divided by the square root of the variance(or standard deviation) of the constraint (SD WL ambiguities or SD NLambiguities, respectively) can be computed. If that ratio exceeds alarge threshold such as three (3), the constraint (SD WL ambiguities orSD NL ambiguities) should be considered problematic. It could be causedby incorrect RTK ambiguity resolution output, for example. The remedyfor this issue is to remove the corresponding problematic constraintsone-by-one from the solution or position estimate of the navigationpositioning estimator 50. An alternative approach is for the navigationpositioning estimator 50 to de-weight those problematic constraint(s) byincreasing their variance, using the ratio as a scaling factor, forexample. Typically, the error checking or error resolution is done in aniterative way.

In step S310 at the mobile receiver 20, the precise positioning module16 or the navigation positioning estimator 50 estimates a preciseposition of the rover 20 based on the converged or fixed narrow-laneambiguities and wide-lane ambiguities. The mobile receiver or thepositioning engine can compute an absolute position of the mobilereceiver based on the relative position and a reference absoluteposition of the mobile receiver 20 at the first measurement time orstored in the data storage device 62. The mobile receiver 20, precisepositioning module 16 or navigation positioning estimator 50 canestimate a precise position of the mobile receiver 20 based on theconverged or fixed narrow-lane ambiguities and wide-lane ambiguitiesbased on the correction data and the backup data. Further, the precisepositioning module 16 can estimate wide-lane (WL) ambiguities,refraction-corrected (RC) ambiguities, and tropospheric bias (e.g.,mobile tropospheric bias) for the mobile receiver 20 based on thecorrection data and backup data. The above steps of FIG. 3 areexecutable or implemented by a data processor 66 of an electronic dataprocessing system 152 of the mobile receiver 20.

In accordance with one embodiment, FIG. 4, which comprises FIG. 4A andFIG. 4B collectively, discloses a method and satellite receiver forrapid determination of precise position by aiding data 30. The method ofFIG. 4 begins in step S101.

In step S100, at regular intervals a mobile receiver 20, a dataprocessor 66 or the precise positioning module 16 (e.g., precise pointpositioning module) determines backup data, such as converged wide-laneambiguities; narrow lane ambiguities or refraction-correctedambiguities; tropospheric delay, reference receive position and rawmeasurements (e.g., phase and code measurements) for a first measurementtime. For example, at regular intervals the mobile receiver 20 or theprecise positioning module 16 determines the backup data for a set ofreceived carrier phase signals, received code signals, or both from aset of satellites of one or more GNSS systems (e.g., GPS, GLONASS,and/or Galileo) in accordance with precise point positioning algorithm.First, the precise positioning module 16 may estimate undifferenced orzero-differenced wide-lane ambiguities. Second, the precise positioningmodule 16 can estimate single-differenced wide-lane (WL) ambiguities.Third, the precise positioning module 16 can use the estimated wide-laneambiguities as constraints or inputs for estimating the narrow-laneambiguities. For instance, at the mobile receiver 20, estimating ofwide-lane (WL) ambiguities, by a predictive filter (e.g., 38, 40), isbased on a LAMBDA (Least-squares AMBiguity Decorrelation Adjustment) ormodified LAMBDA procedure to prepare for determination of thenarrow-lane ambiguities in integer form (e.g., in accordance with a BestInteger Equivariant (BIE) or modified BIE algorithm). The BIE is aambiguity resolution or ambiguity fixing technique that can minimize themean squared error of the integer ambiguities or the real part of thefloating solution.

In step S102, the mobile receiver 20, the backup recovery module 46, orthe data processor 66 stores or records the backup data on a datastorage device 62 (e.g., nonvolatile random access memory). For example,the backup/recovery module 46 or the navigation positioning estimator 50stores backup data determined in step S100.

In step S104, the mobile receiver 20, cycle slip detector 59, or dataprocessor 66 detects: (1) loss, interruption, disruption, interference,or corruption (e.g., time of loss) of reception of satellite signals bythe mobile receiver 20 or of the correction signal, which carriescorrection data (108), by the correction wireless device 26, or loss orinterruption of electrical power to the mobile receiver 20 and/or thecorrection wireless device 26, and (2) restoration (e.g., time ofrestoration) of reception (e.g., reliable reception or sufficient signalquality) of lost satellite signals, correction signal or electricalpower (e.g., direct current power to mobile receiver 20 from a vehicle)after the signal loss, interruption, disruption, interference,corruption or electrical power loss. The reliable or sufficient signalquality of the received satellite signals, the received correctionsignal, or both may be determined by any of the following: (a) receivedsignal strength indicator of the received signal, (b) signal qualityindicator of the received signal, (c) bit error rate of informationencoded (e.g., navigation data, pseudo-range data, or (GPS)coarse-acquisition code data encoded on GNSS carriers) on the receivedsignal, (d) dilution of precision or other figure of merit for thereceived signal, or (e) loss-of-lock, cycle slips or repeated cycleslips detected by the cycle slip detector 59 during a time interval forthe received satellite signal.

In step S106, the mobile receiver 20, backup recover module 46, or dataprocessor 66 determines whether or not the detected restoration occurswithin a maximum recovery time period from the first measurement time orafter a detection time (e.g., first detection time) of the loss,interruption, disruption, corruption or interference with the receivedsatellite signals, the correction signal or loss of electrical power. Ifthe detected restoration occurs within the maximum recovery time periodfrom the first measurement time or the detection time, the methodcontinues with step S108. However, if the detected restoration does notoccur within the maximum recovery time period, the method returns tostep S100 because the backup data is regarded as too stale orinsufficiently reliable to be used for the rapid recovery process.

In step S108, the mobile receiver 20, backup recovery module 46 or thedata processor 66 retrieves backup data from the data storage device 62for the mobile receiver 20 at the first measurement time.

In step S112, the mobile receiver 20 measures carrier phase of receivedsatellite signals at a second measurement time. For example, the secondmeasurement time refers to a measurement time that follows the firstmeasurement time within the maximum recover time period. In other words,a second epoch, current epoch, or next epoch, follows the first epochwithin the maximum recovery time period.

In step S114, the mobile receiver 20, the relative positioning module18, or the real-time kinematic (RTK) filter 48 determines a doubledifference between carrier phase measurement of the mobile receiver 20between the first measurement time and the second measurement time andbetween a pair of satellites. For example, the mobile receiver 20, therelative positioning module 18, or the real-time kinematic (RTK) filter48 determines a double difference between carrier phase measurements ofthe mobile receiver 20 between the first measurement time and the secondmeasurement time and between a pair of satellites to cancel out bias(e.g., code bias, phase bias, and clock bias) common between a pair ofsatellites and the rover 20 and the reference receiver 130. In stepS114, the mobile receiver 20, relative positioning module 18, ornavigation positioning estimator 50 determines a set of doubledifferences with respect to multiple respective pairs of satellitesobserved at the mobile receiver.

In an alternate embodiment, step S114 and step S116 may be combined orexecuted simultaneously.

In step 116, the mobile receiver 20, relative positioning module 18, orthe real-time kinematic (RTK) filter 48 estimates double-differencedL1/L2 integer ambiguities (e.g., double difference L1/L2 RTK integerambiguities) consistent with the determined double differences (e.g., ofstep S116). In this document, double-differenced L1/L2 integerambiguities or real-time kinematic (RTK) double-differenced L1/L2ambiguities means any of the following: DD L1 RTK integer ambiguities,DD L2 RTK integer ambiguities or both. Step S116 may be carried out inaccordance with various techniques, which may be applied separately andcumulatively.

Under a first technique, the mobile receiver 20, relative positioningmodule 18 or real-time kinematic (RTK) module estimatesdouble-differenced L1/L2 integer ambiguities, such as DD L1 RTK integerambiguities, DD L2 RTK integer ambiguities or both, by minimizing theerror associated with a least squares equation to search for optimal oracceptable integer ambiguity solutions among candidate integer ambiguitysolutions.

Under a second technique, the mobile receiver 20, relative positioningmodule 18 or real-time kinematic (RTK) module estimatesdouble-differenced L1/L2 integer ambiguities by a LAMBDA (Least-squaresAMBiguity Decorrelation Adjustment) or modified LAMBDA method. Forinstance, the error minimization of the least squares equation fordecorrelated ambiguities is carried out over a search region determinedby a variance and covariance matrix of the ambiguities; floatingambiguity estimates and associated variance/co-variance matrices can beused as inputs to the LAMBDA process, where the output is integerambiguity estimates.

Under a third technique, the mobile receiver 20, relative positioningmodule 18 or real-time kinematic (RTK) module estimatesdouble-differenced L1/L2 integer ambiguities and ionosphere delay biasper satellite.

In step S118, the mobile receiver 20, relative positioning module 18, orthe real-time kinematic (RTK) module determine the relative positionvector between the mobile receiver 20 at the first measurement time andthe second measurement time in accordance with the estimateddouble-differenced L1/L2 (e.g., RTK) integer ambiguities, such as DD L1RTK integer ambiguities, DD L2 RTK integer ambiguities or both. Forexample, the double-differenced L1/L2 integer ambiguities can beresolved in accordance with Equations 24 and 25. In practice, therelative positioning module 18 can also use the pseudo-range equations(Equations 22 and 23) as constraints to resolve the DD L1 RTK integerambiguities and DD L2 RTK integer ambiguities.

In step S120, the mobile receiver 20, relative positioning module 18, orthe real-time kinematic (RTK) module 48 determine the double difference,wide-lane WL RTK integer ambiguity and the refraction-corrected RTKinteger ambiguity between satellites (e.g., in reception range of thereference receiver 130) based on the estimated double-difference L1/L2(e.g., RTK) integer ambiguities as constraints and based on atmosphericor tropospheric delay bias in accordance with atmospheric models.

In step S122, a mobile receiver 20, its data processor 66 or its precisepoint positioning module 16 applies backup data (from the data storagedevice 62), such as converged wide-lane ambiguities; narrow laneambiguities or refraction-corrected ambiguities; tropospheric delay,reference receive position and raw measurements (e.g., phase and codemeasurements) to speed up predictive filter convergence in the precisepoint positioning module 16 of the rover 20. Step S122 may be carriedout in accordance with various techniques, which may be appliedseparately or cumulatively.

Under a first technique, in step S122, a mobile receiver 20, its dataprocessor 66 or its precise point positioning module 16 (e.g.,backup/recovery module 46) applies the backup data, such as the resolveddouble difference wide lane ambiguities (e.g., from step S120), theresolved refraction-corrected RTK integer ambiguity (e.g., from stepS120), the relative position vector or relative position (e.g., fromstep S118) to be used as constraint data for one or more predictivefilters (e.g., wide-lane filter 40, narrow-lane filter 44) associatedwith the precise point positioning module.

Under a second technique for carrying out step S108, a mobile receiver20, its data processor 66 or its precise point positioning module 16applies backup data, such as variance and covariance of the ambiguitiesas constraint data for one or more predictive filters (e.g., wide-lanefilter 40, narrow-lane filter 44) associated with the precise pointpositioning module.

In step S124, a mobile receiver 20, its data processor 66 or its precisepoint positioning module 16 (e.g., backup/recovery module 46) determinesor recovers a rover wide-lane single-differenced (SD) ambiguity based onone or more of the following: (1) single-difference wide-lane integerambiguity fixed or resolved at the mobile receiver 20 at a firstmeasurement time (e.g., t₁), and (2) resolved or fixed double-difference(DD) wide-lane ambiguity (e.g., RTK) associated with the firstmeasurement time and the second measurement time (e.g.,satellite-differenced DD wide-lane RTK ambiguities with respect to apair of satellites and time-differenced with respect to differentmeasurement times).

In optional step S126, a mobile receiver 20, its data processor 66 orits precise point positioning module determines: (1) a variance andcovariance matrix for relative position of mobile receiver 20 to use asconstraint data for one or more predictive filters, such as the narrowlane predictive filter associated with the precise point positioningmodule, (2) a single-difference narrow lane ambiguity based on thesingle difference wide-lane integer ambiguity, or based on thecombination of the variance and covariance matrix for relative positionand the single difference wide-lane integer ambiguity. For example, thenavigation positioning estimator 50 can use the variance to determinestandard deviation of the resolved ambiguities or standard deviation ofthe position estimates to measure the quality of the resolvedambiguities and position estimates. Further, the navigation positioningestimator 50 can use the variance or determined standard deviationsdecide whether to eliminate or reduce weighting of certain less reliable(or more variable) carrier phase measurements from certain satellitesfrom the final position estimate or solution. Step S126 is optional asindicated by the dashed lines.

In step S128, the mobile receiver 20, its data processor 66 or itsprecise point positioning module 16 (e.g., backup recover module) maydetermine an estimated rover position based on resolution of the narrowlane ambiguities associated with the received carrier phase signals atthe mobile receiver 20. For example, the precise point positioningmodule 16 may ignore, discount or reduce the weighting of carrier phasemeasurements that are determined to be unreliable in optional step S126to arrive at the estimated rover position.

In an alternate embodiment, step S128 and step S130 may be combined orexecuted simultaneously.

In step S130, the mobile receiver 20, its data processor 66 or itsprecise point positioning module 16 may determine refraction-corrected,single-difference ambiguity for the mobile receiver 20 based on theconverged refraction-corrected ambiguity of the mobile receiver 20 atthe first measurement time and the refraction-correcteddouble-difference ambiguity (RTK). For example, the tropospheric delaybias of the mobile receiver 20 at the first measurement timeapproximately equals (e.g., with a tolerance plus or minus five percent)the tropospheric bias of the mobile receiver 20 at the secondmeasurement time, where the first measurement time and the secondmeasurement time are within a maximum time period.

FIG. 5 illustrates an optional error checking procedure or method thatmay be implemented in conjunction with the method of FIG. 3, such asafter an iteration of the method or after step S308 or as part of stepS126 of FIG. 4B.

In step S500, the navigation positioning estimator 50, the precisepositioning module 16, or an error detection module therein, determinespost-fit residuals (for the SD WL ambiguities or SD NL ambiguities,respectively) after applying one or more constraints (SD WL ambiguitiesor SD NL ambiguities, respectively) to single-difference wide-lane,narrow-lane equations, or atmospheric (e.g. tropospheric) model.

In step S502, the navigation positioning estimator 50, the precisepositioning module 16 or the error detection module determines ratio ofpost-fit residual, or its standard deviation, (for the SD WL ambiguitiesor SD NL ambiguities, respectively) divided by the square root ofvariance (i.e., standard deviation) of the constraint (SD WL ambiguitiesor SD NL ambiguities, respectively). The square root of the variance isalso referred to as the standard deviation of a variable (e.g., for anormal distribution).

In step S504, the navigation positioning estimator 50, the precisepositioning module 16 or the error detection module determines whetheror not the ratio greater than a threshold. The threshold may comprise aninteger (e.g., 2 or 3) or a real-valued number that is based onempirical data, field testing, a service level for the correction dataor the particular mobile receiver 20, or derived from equations thatdepend upon the current location or geographic zone of a mobile receiver20. If the navigation positioning estimator 50, the precise positioningmodule 16 or the error detection module determines that the ratio isgreater than the threshold, then the method continues with step S506.However, if the navigation positioning estimator 50, the precisepositioning module 16 or the error detection module determines that theratio is not greater than the threshold (or is equal to the threshold),then the method continues with step S508.

In step S506, the precise positioning module 16 or its components testeach constraint by removing the constraint or de-weighting theconstraint from relevant single-difference wide-lane equations,narrow-lane equation(s) or atmospheric models. For example, the precisepositioning module 16 or the navigation positioning estimator 50 mayeliminate, ignore, discount or reduce the weighting of carrier phasemeasurements that are determined to be unreliable to arrive at theestimated rover position or solution in any of the methods or proceduresset forth in this document.

In step S508, the precise positioning module 16 or its componentsdetermine that the constraint value is okay or within an acceptabletolerance (e.g., of expected value or error minimization process). Forexample, the precise positioning module 16 or the navigation positioningestimator 50 may include or maintain the weighting of carrier phasemeasurements that are determined to be unreliable to arrive at theestimated rover position or solution in any of the methods or proceduresset forth in this document.

The foregoing description, for purpose of explanation, has beendescribed with reference to specific embodiments. However, theillustrative discussions above are not intended to be exhaustive or tolimit the invention to the precise forms disclosed. Many modificationsand variations are possible in view of the above teachings. Theembodiments were chosen and described in order to best explain theprinciples of the invention and its practical applications, to therebyenable others skilled in the art to best utilize the invention andvarious embodiments with various modifications as are suited to theparticular use contemplated.

The following is claimed:
 1. A method for providing or rapidlyrecovering an estimated position of a mobile receiver, the methodcomprising execution of the following steps at or by the mobilereceiver: receiving a set of one or more carrier satellite signals and acorrection signal encoded with correction data that is related to theone or more carrier satellite signals; measuring the carrier phase ofone or more received satellite signals a first measurement time;estimating a wide-lane ambiguity and narrow-lane ambiguity in themeasured carrier phase of the one or more received satellite signals forthe first measurement time and estimating tropospheric bias for one ormore of the carrier satellite signals; storing, at regular timeintervals for the first measurement time, backup data comprising a setof the following post-convergence or resolved values: the estimatedwide-lane ambiguities, the estimated narrow-lane ambiguities, theestimated tropospheric delay bias, raw measured carrier phase of thereceived satellite signals, and corresponding estimated receiverpositions; detecting a loss of lock on the measured carrier phaseassociated with loss or lack of reception of one or more of the carriersignals for a loss time period; after the detected loss of lock once atleast some carrier phase signals are reacquired, measuring the carrierphase of one or more received satellite signals at a second measurementtime; retrieving or reading the backup data and applying the backup datato a real-time kinematic (RTK) filter to provide a relative positionvector between the mobile receiver at the first measurement time and themobile receiver at the second measurement time and to provide recoverydata associated with a satellite-differenced double-differenceestimation for the mobile receiver between the first measurement timeand the second measurement time; applying the relative position vector,the backup data, the recovery data from the RTK filter, and thecorrection data with precise clock and orbit information on thesatellite signals, as inputs, constraints, or both for convergence orresolution of one or more predictive filters on wide-lane andnarrow-lane ambiguities in accordance with a precise positioningalgorithm; estimating a precise position of the mobile receiver based onthe resolved narrow-lane ambiguities and wide-lane ambiguities that arein a converged state or fixed state, where the above steps areexecutable or implemented by a data processor of an electronic dataprocessing system of the mobile receiver; wherein the mobile receiverposition at the first measurement time comprises a precise pointpositioning position of the mobile receiver at the first measurementtime, and wherein the recovery data includes one or more of thefollowing: RTK double-difference (DD) wide-lane integer ambiguity,∇ΔN_(WL) ^(ij)(RTK); RTK fixed refraction-corrected (RC)double-difference (DD) ambiguity, ∇ΔN_(RC) ^(ij)(RTK); and the relativeposition between the reference receiver and the mobile receiver, and thevariance/co-variance of the relative position.
 2. The method accordingto claim 1 wherein the backup data further comprises one or more of thefollowing estimated at the mobile receiver at the first measurementtime: fixed wide-lane ambiguities that are fixed to integer ambiguityvalues, estimated refraction-corrected ambiguities, raw measurements, ortropospheric delay at zenith direction including an a priori model. 3.The method according to claim 1 wherein the recovery data comprisesL1/L2 fixed double-difference (DD) ambiguities from the RTK filter atthe mobile receiver based on raw phase measurements at the mobilereceiver at the first measurement time and the second measurement timefor a pair of satellites.
 4. The method according to claim 1 whereinapplying the backup data to real-time kinematic (RTK) filter furthercomprises: estimating, by the RTK filter, a relative position or therelative position vector of mobile receiver between the firstmeasurement time and the second measurement time based on a set ofreal-time kinematic (RTK) algorithms to resolve the L1/L2double-differenced (DD) fixed integer values (N₁, N₂) at an L1 frequencyof the carrier satellite signals and a pair of satellites per GlobalNavigation Satellite System (GNSS) system and between an L2 frequency ofthe carrier satellite signals and the same pair of satellites per GNSSsystem.
 5. The method according to claim 1 wherein the resolved valuesof the wide-lane ambiguities comprise pulled-in or converged wide-laneambiguities and wherein the resolved values of the narrow-laneambiguities comprise pulled-in or converged narrow-lane ambiguities fromone or more Global Navigation Satellite System (GNSS) satellites'carrier signals at the mobile receiver.
 6. The method according to claim1 wherein estimating the precise position of the mobile receiver furthercomprises estimating refraction-corrected (RC) ambiguities to theconverged state and tropospheric bias for the mobile receiver based onthe backup data and the correction data.
 7. The method according toclaim 1 further comprising: at the mobile receiver, estimating ofwide-lane (WL) ambiguities, by a predictive filter or wide-lane filter,is based on a LAMBDA or modified LAMBDA procedure to prepare fordetermination of the narrow-lane ambiguities in integer form.
 8. Themethod according to claim 1 wherein further comprising: determiningwhether a reacquisition or detected restoration of the reception of thecarrier phase signals is within a maximum recovery time period prior toapplying the real-time kinematic (RTK) filter to resolve ambiguitiesassociated with double-difference carrier phase measurements.
 9. Themethod according to claim 1 wherein, at the mobile receiver at the firstmeasurement time, refraction-corrected ambiguities, (N_(RC)) forsatellite j are determined based on a Kalman filter in accordance withthe following equations after resolving the wide-lane ambiguities:$\mspace{20mu}{P_{RC}^{j} = {{{\frac{f_{1}^{2}}{f_{1}^{2} - f_{2}^{2}}P_{1}^{j}} - {\frac{f_{2}^{2}}{f_{1}^{2} - f_{2}^{2}}P_{2}^{j}}} = {\rho^{j} + \tau_{r} + \tau^{j} + T + {ɛ_{P_{RC}}^{j}\mspace{14mu}{and}}}}}$$L_{RC}^{j} = {{{\frac{f_{1}^{2}}{f_{1}^{2} - f_{2}^{2}}L_{1}^{j}} - {\frac{f_{2}^{2}}{f_{1}^{2} - f_{2}^{2}}L_{2}^{j}}} = {\rho^{j} + \tau_{r} + b_{NL} + \tau^{j} + B_{NL}^{j} + T + {( {N_{RC}^{j} + W^{j} + w} )\lambda_{NL}} + ɛ_{L_{RC}}^{j}}}$where: P_(RC) ^(j) is the refraction-corrected phase code for satellitej; P₁ ^(j) is the measured phase code on the L1 frequency for satellitej; P₂ ^(j) is the measured phase code on the L2 frequency for satellitej; ε_(P) _(RC) ^(j) is the RC phase code measurement error for satellitej including white noise, multipath and remaining un-modeled errors; f₁is the L1 carrier frequency and f₁ is the L2 carrier frequency of thereceived satellite signals; L₁ ^(j) is the measured carrier phase forthe L1 carrier frequency transmitted from satellite j; L₂ ^(j) is themeasured carrier phase for the L2 carrier frequency transmitted fromsatellite j; ρ^(j) is the geometric distance between the satellite jphase center and the receiver phase center including satellite orbitalcorrection, receiver tide displacement and earth rotation correction;τ_(r) is the receiver r clock bias or error for a given GNSS system;τ^(j) is the satellite clock error; b_(NL) is the receiver narrow-lanephase bias (one per receiver and each constellation for all visiblesatellites), B_(NL) ^(j) is the satellite j narrow lane phase bias (oneper satellite), which is a RC combination of the L1 satellite phase biasand the L2 satellite phase bias; T is the tropospheric delay, and isdivided into a dry component T_(dry) and a wet component T_(wet); W^(j)and w are phase windup errors for both satellite j and receiver, incycles, respectively, which can be corrected with models; N_(RC) ^(j) isthe refraction-corrected (RC) carrier phase ambiguity term;$\lambda_{NL} = \frac{c}{f_{1} + f_{2}}$ is the narrow lane wavelength;and ε_(L) _(RC) ^(j) is the RC phase measurement error for satellite jincluding white noise, multipath and remaining un-modeled errors. 10.The method according to claim 1 wherein the resolved double-differenced(DD) L1/L2 integer ambiguities are determined in accordance with thefollowing equations: $\begin{matrix}{{{\nabla\Delta}\; L_{1}^{ij}} = {{\nabla{\Delta\rho}^{ij}} - {{\nabla\Delta}\; I^{ij}} + {{\nabla N_{1}^{i}}\lambda_{1}^{i}} - {\Delta\; N_{1}^{j}\lambda_{1}^{j}} + {\nabla{\Delta ɛ}_{L_{1}}^{ij}}}} \\{{{\nabla\Delta}\; L_{2}^{ij}} = {{\nabla{\Delta\rho}^{ij}} - {\frac{f_{1}^{2}}{f_{2}^{2}}{\nabla\Delta}\; I^{ij}} + {\Delta\; N_{2}^{i}\lambda_{2}^{i}} - {\Delta\; N_{2}^{j}\lambda_{2}^{j}} + {\nabla{\Delta ɛ}_{L_{2}}^{ij}}}}\end{matrix}$ where: ∇ΔL₁ ^(ij) is the double-difference L1 carrierphase measurements with respect to satellite i and j, the mobilereceiver between the first measurement time and the second measurementtime; ∇Δρ^(ij) is the double-difference geometric distance between thesatellite j phase center and a receiver phase center and betweensatellite i phase center and the receiver phase center, includingsatellite orbital correction, receiver tide displacement and earthrotation correction; ∇ΔI^(ij) is the double-difference ionosphere errorfor a given satellite j and satellite i; ∇N₁ ^(i)λ₁ ^(i) is thesingle-differenced integer ambiguity for satellite i multiplied by thewavelength for the L1 carrier from satellite i; ∇N₁ ^(j)λ₁ ^(j) is thesingle-differenced integer ambiguity for satellite j multiplied by thewavelength for the L1 carrier from satellite j; ∇Δε_(L) ₁ ^(ij) is thedouble-difference phase measurement error for satellite j and satellitei including white noise, multipath and remaining un-modeled errors withrespect to the L1 frequency; ∇ΔL₂ ^(ij) is the double-difference L2carrier phase measurements with respect to satellite i and j and themobile receiver between the first measurement time and the secondmeasurement time; ΔN₂ ^(i)λ₂ ^(i) is the single-differenced integerambiguity for satellite i multiplied by the wavelength for the L2carrier from satellite i; ΔN₂ ^(j)λ₂ ^(j) is the single-differencedinteger ambiguity for satellite j multiplied by the wavelength for theL2 carrier from satellite j; ∇Δε_(L) ₂ ^(ij) is the double-differencephase measurement error for satellite j and satellite i including whitenoise, multipath and remaining un-modeled errors with respect to the L2frequency; and f₁ is the L1 carrier frequency and f2 is the L2 carrierfrequency of the received satellite signals.
 11. The method accordingclaim 10 wherein after resolution of the double-difference L1/L2ambiguities in claim 10, RTK wide-lane ambiguities are determined inaccordance with the following equation:∇ΔN _(WL) ^(ij)(RTK)=∇ΔN ₁ ^(ij) −∇ΔN ₂ ^(ij) where: ∇ΔN_(WL) ^(ij)(RTK)is the RTK double-difference wide-lane ambiguity for satellites i and jwith respect to the mobile receiver at the first measurement time andthe second measurement time; ∇ΔN₁ ^(ij) is the double-difference L1wide-lane ambiguity for satellites i and j with respect to the mobilereceiver at the first measurement time and the second measurement time;and ∇ΔN₂ ^(ij) is the double-difference L2 wide-lane ambiguity forsatellites i and j with respect to the mobile receiver at the firstmeasurement time and the second measurement time.
 12. The methodaccording to claim 1 further comprising providing aiding data comprisingrefraction-corrected ambiguity between satellite i and j can be derivedin accordance with the following equation:${{\nabla\Delta}\;{N_{RC}^{ij}({RTK})}} = {{\frac{f_{1}^{2}}{f_{1}^{2} - f_{2}^{2}}{\nabla\Delta}\; N_{1}^{ij}} - {\frac{f_{2}^{2}}{f_{1}^{2} - f_{2}^{2}}{\nabla\Delta}\; N_{2}^{ij}}}$where: ∇ΔN_(RC) ^(ij) (RTK) is the RTK refraction-correcteddouble-difference wide-lane ambiguity for satellites i and j withrespect to the mobile receiver at the first measurement time and thesecond measurement time; ∇ΔN₁ ^(ij) is the double-difference L1wide-lane ambiguity for satellites i and j with respect to the mobilereceiver at the first measurement time and the second measurement time;∇ΔN₂ ^(ij) is the double-difference L2 wide-lane ambiguity forsatellites i and j with respect to the mobile receiver at the firstmeasurement time and the second measurement time; and f₁ is thefrequency of the L1 carrier phase signal and f₂ is the frequency of theL2 carrier phase signal.
 13. The method according to claim 1 wherein thefollowing parameters of double-difference (DD) wide-lane integerambiguity, the fixed refraction-corrected ambiguity, the relativeposition ΔX and their variance/co-variance are used as additionalconstraints to speed up a filter convergence process of the one or morepredictive filters at the mobile receiver.
 14. The method according toclaim 1 further comprising: using RTK double-difference wide-laneambiguities ∇ΔN_(WL) ^(ij)(RTK) for satellites i and j (with respect tothe mobile receiver at the first measurement time and the secondmeasurement time) and the single-difference wide lane ambiguity, ∇N_(WL)^(ij)(t₁), at the mobile receiver at the first measurement time toderive or estimate the respective single-difference ambiguities, ∇N_(WL)^(ij)(t₂), at the mobile receiver at a second measurement time inaccordance with the following equation:∇N _(WL) ^(ij)(t ₂)=∇ΔN _(WL) ^(ij)(RTK)+∇N _(WL) ^(ij)(t ₁).
 15. Themethod according to claim 1 further comprising: using an estimatedposition X(t₂) of the mobile receiver and a corresponding covariancematrix Q_(XYZ)(t₂) for the estimated position of the mobile receiver atthe second measurement time as virtual measurements or constraints to beapplied to a narrow-lane filter as the one or more predictive filters inaccordance with the following equations:X(t ₂)=ΔX _(RTK) −X(t ₁) andQ _(XYZ)(t ₂)=Q _(ΔXYZ(RTK)) ±Q _(XYZ)(t ₁) where: ΔX_(RTK) is arelative change in position of the mobile receiver that is observed bythe RTK filter between the first measurement time, t₁, and the secondmeasurement time, t₂; X(t₁) is the estimated position of the mobilereceiver at the first measurement time, t₁; Q_(ΔXYZ(RTK)) is the changein the covariance matrix associated with the relative change in positionΔX_(RTK); and Q_(XYZ) (t₁) is the covariance matrix for the estimatedposition of the mobile receiver at the first measurement time.
 16. Themethod according to claim 1 further comprising: using previouslyestimated troposphere bias T(t₂) and corresponding covariance matrixQ_(T_(t₂)) as virtual measurements or constraints to be applied to anarrow-lane filter as the one or more predictive filters in accordancewith the following equations: $\begin{matrix}{{T( t_{2} )} \approx {{T( t_{1} )}\mspace{14mu}{and}}} \\{Q_{T_{t_{2}}} = {Q_{T_{t_{1}}} + {{\Delta\; t} \star q_{Trop}^{Temporal}} + {{{\Delta\; X_{RTK}}} \star q_{Trop}^{Spatial}}}}\end{matrix}$ where: T(t₂) is the estimated tropospheric delay for themobile receiver at the second measurement time, t₂; and T(t₁) is theestimated tropospheric delay for the mobile receiver (e.g., by theatmospheric bias estimator 42) at the first measurement time, t₁;Q_(T_(t₂)) is the variance of estimated tropospheric delay at the secondmeasurement time, t₂; q_(Trop) ^(Spatial) is a spatial correlationfactor for inflating the covariance over spatial displacement of themobile receiver; q_(Trop) ^(Temporal) is a temporal correlation factorfor inflating the covariance over time; |ΔX_(RTK)| is traveled distanceof the mobile receiver from the first measurement time, t₁, to thesecond measurement time, t₂, and Δt=t₂−t₁ or the time difference betweenthe first measurement time and the second measurement time.
 17. Themethod according to claim 1 further comprising: using RTKdouble-difference refection-corrected ambiguities ∇ΔN_(RC) ^(ij)(RTK)for satellites i and j (with respect to the mobile receiver at the firstmeasurement time and the second measurement time) and thesingle-difference refraction-corrected ambiguity, ∇ΔN_(RC) ^(ij)(t₁), atthe mobile receiver at the first measurement time to derive or estimatethe respective single-difference refection-corrected ambiguities,∇ΔN_(RC) ^(ij)(t₂), at the mobile receiver at a second measurement timein accordance with the following equation:∇N _(RC) ^(ij)(t ₂)=∇ΔN _(RC) ^(ij)(RTK)+∇N _(RC) ^(ij)(t ₁).
 18. Amobile receiver for quickly determining a precise position based oncorrection data received from a correction data source, the mobilereceiver comprising: a receiver module for receiving a set of one ormore satellite signals; a measurement module for measuring the carrierphase of one or more received satellite signals for a first measurementtime; a correction wireless device for receiving a correction signalrelated to the set of one or more satellite signals; an estimator forestimating a wide-lane ambiguity and narrow-lane ambiguity in themeasured carrier phase of the one or more received satellite signals forthe first measurement time and estimating tropospheric bias for one ormore of the carrier satellite signals; a data storage device forstoring, at regular time intervals for the first measurement time,backup data comprising a set of the following post-convergence orresolved values: the estimated wide-lane ambiguities, the estimatednarrow-lane ambiguities, the estimated tropospheric delay bias, rawmeasured carrier phase of the received satellite signals, and one ormore corresponding estimated receiver positions; a detector fordetecting a loss of lock on the measured carrier phase associated withloss or lack of reception of one or more of the carrier satellitesignals for a loss time period; after the detected loss of lock once atleast some carrier phase signals are reacquired, the measurement moduleadapted to measure the carrier phase of one or more received satellitesignals at a second measurement time; a real-time kinematic (RTK) filterfor using the backup data to estimate a relative position vector betweenthe mobile receiver at the first measurement time and the mobilereceiver at the second measurement time and to provide recovery dataassociated with a satellite-differenced double-difference estimation forthe mobile receiver between the first measurement time and the secondmeasurement time; a navigation positioning estimator for applying therelative position vector, the backup data, the recovery data from theRTK filter, and the correction data with precise clock and orbitinformation on the satellite signals, as inputs, constraints, or bothfor convergence or resolution of wide-lane and narrow-lane ambiguitiesin accordance with a precise positioning algorithm; and the navigationpositioning estimator adapted to estimate a precise position of themobile receiver based on the resolved narrow-lane ambiguities andwide-lane ambiguities that are in a converged state or fixed state,where the above is implemented by a data processor of an electronic dataprocessing system of the mobile receiver; wherein the mobile receiverposition at the first measurement time comprises a precise pointpositioning position of the mobile receiver at the first measurementtime, and wherein the recovery data includes one or more of thefollowing: RTK double-difference (DD) wide-lane integer ambiguity,∇Δ_(WL) ^(ij) (RTK); RTK fixed refraction-corrected (RC)double-difference (DD) ambiguity, ∇Δ_(RC) ^(ij) (RTK); and the relativeposition between the reference receiver and the mobile receiver; and thevariance/co-variance of the relative position.
 19. The mobile receiveraccording to claim 18 wherein the correction wireless device comprises asatellite receiver for receiving the correction signals in the L-band.20. The mobile receiver according to claim 18 wherein the one or moreestimated receiver positions comprise a precise point positioningreference position of the mobile receiver that is stationary or mobileat the first measurement time.
 21. The mobile receiver according toclaim 18 wherein the backup data further comprises one or more of thefollowing estimated at the reference receiver: resolved wide-laneambiguities, estimated refraction-corrected ambiguities, rawmeasurements, and tropospheric delay at zenith direction associated withan a priori model.
 22. The mobile receiver according to claim 18 whereinthe recovery data comprises the L1/L2 fixed DD ambiguities from the RTKfilter at the mobile receiver.
 23. The mobile receiver according toclaim 18 wherein the real-time kinematic (RTK) filter is adapted toestimate a relative position or relative position vector between amobile receiver at the first measurement time and the second measurementtime based on a set of real-time kinematic (RTK) algorithms to resolvethe L1/L2 double-differenced (DD) fixed integer values (N₁, N₂) at an L1frequency of the carrier satellite signals and a pair of satellites perGlobal Navigation Satellite System (GNSS) system and between an L2frequency of the carrier satellite signals and the same pair ofsatellites per GNSS system.
 24. An electronic data processing system forproviding rapid position recovery of a rover navigation receiver afterpartial or full loss of received satellite signals, the systemcomprising: a receiver module for receiving a set of one or moresatellite signals; a correction wireless device for receiving acorrection signal related to the set of one or more satellite signals; ameasurement module for measuring the carrier phase of each one of thesatellite signals in the set with respect to a local reference carriersignal; an estimation module for estimating a wide-lane ambiguity andnarrow-lane ambiguity in the measured carrier phase and troposphericbias for each carrier signal in the set; a data storage device forstoring, at regular time intervals, backup data comprisingpost-convergence values of the estimated wide-lane ambiguity, theestimated narrow-lane ambiguity, raw code measurements and raw carrierphase measurements, the tropospheric bias and a corresponding referenceposition for a first measurement time; a detector for detecting a lossof lock on the measured carrier phase associated with loss or lack ofreception of one or more of the satellite signals for a loss timeperiod; and upon the detected loss of lock after some carrier phasesignals are reacquired, a recovery module for recovering an accuratecurrent position estimate, the tropospheric bias, the current wide-laneambiguity, the current narrow-lane ambiguity based on the stored backupdata and observed measurements of the carrier phase of the one or morereceived satellite signals at a second measurement time to estimate arelative position vector between the mobile receiver at the firstmeasurement time and the mobile receiver at the second measurement time.25. The electronic data processing system according to claim 24 whereinthe recovery module is arranged to process a double-differencedobservable for each satellite for the first measurement time, at a lastfirst epoch, prior to the loss of lock and for the second measurementtime, at a current epoch during a recovery of the accurate positionestimate to precisely estimate the relative position estimate withrespect to the last stored reference position.
 26. The data processingsystem according to claim 25 wherein the double-differenced observablecomprises carrier phase measurements for L1 frequency or L2 frequency ofthe received satellite signals, or code phase measurements for L1frequency or L2 frequency of received satellite signals.
 27. The dataprocessing system according to claim 24 wherein the recovery module isarranged to conduct ambiguity resolution to recover wide-lane ambiguitychanges and narrow-lane ambiguity changes during a re-convergence timeperiod on the fixed L1 and L2 ambiguities in a real-time kinematicfilter.